 This is an informal talk on sporadic simple groups that I gave to the Archimedians, the Cambridge Undergraduate Mathematical Society. So I'll start just by reminding you what a group is. So group theory is really the study of symmetry. And symmetry is very close related to whether objects look beautiful or not. So here are some examples of objects, some of which are symmetric and some of which are not. So down here there's some sort of crab, one of whose claws is much bigger than the other. So it's unsymmetric and this is kind of a bit disconcerting. On the right here we have various symmetric objects such as flowers and diamonds, all of which sort of look rather pretty. On the left we have a potato which doesn't look very pretty. And one of the reasons it doesn't look particularly pretty is it's simply not symmetric. So symmetry is sort of a mathematical study of beauty in some sense. And then we can ask what a group is. Well a group is just the collection of symmetries of an object, where a symmetry is just a way of mapping the object to itself in a way that preserves structure. So for example suppose we take a tetrahedron, we can ask how many symmetries does it have? Well it's got four faces and we can map any of these four faces to the bottom. And once we've put one of the four faces on the bottom there are three ways to rotate it preserving that face. So the total number of symmetries is four times three which is 12. Similarly a cube has six times four symmetries because it has six faces each with four sides. Unless it's a Rubik's Cube and unless it's got, in which case it's got vastly more symmetries but never mind. And two properties of a group we're going to use is first of all the order of a group which is just the number of elements. So the symmetries of a cube has maybe 24. There's order 24 or possibly 48 if you allow reflections as well. And the order of an element of a group is the number of times you need to multiply this element by itself to get back to where you started. For example rotating a tetrahedron by third of a revolution is a group element of order three. And a basic problem is to classify all possible groups. Well obviously this is just too complicated in general but we can classify all groups of a given order. And for example there are two possible groups of order four. So on the top half of this slide all these objects have the same automorphism group which has one symmetry of order one and three symmetries of order two. On the other hand all the objects on the bottom also have a symmetry group of order four. But in this case there are two elements of order four which are usually rotations by a quarter of a revolution. So there are two possible groups of order four up to isomorphism. Isomorphism means two groups are isomorphic if they're really the same up to relabeling the elements. So as an exercise you can try and figure out what is the group of symmetries of a tennis ball. There are four symmetries so which group is it? Is it the one with three elements of order two or is it the one with two elements of order four? A slightly more complicated question is to ask what is the symmetry group of an apple. And you might think at first that it's got an infinite number of symmetries because you can just sort of rotate it. If you slice through an apple you'll find it's got this nice pentagonal arrangement of seeds in the middle. And its symmetry group is order five or order ten depending on whether you allow reflections. If you allow reflections you can see it's got two rotations by fifth of a revolution, two rotations by two fifths of a revolution and five reflections in for instance you can reflect in this line here and there are five similar lines. So there are four types of symmetry of the apple and these are called its conjugacy classes. So roughly speaking two symmetries are said to be conjugate or in the same conjugacy class if they kind of look the same. I mean a revolution by one fifth of a revolution clockwise looks the same as a rotation by one fifth of a revolution counterclockwise. So these are in the same conjugacy class. There's another group of order ten which just consists of all rotations of a decagon and you can see these groups are different because this rotations of a decagon has symmetries of order ten but the apple doesn't. So we can try and classify all groups of small order and for order less than about ten this isn't terribly difficult. So here's a list of them. There's always at least one group of any given order because we can take a if we want a group of order seven say all we do is we take a regular seven sided polygon and look at the group of its rotations. That's called a cyclic group because the elements of the group are a bit like rotating a circle. Then we get two groups of order four as we've mentioned and we get two groups of order six because as well as rotations of a hexagon you can look at all rotations and reflections of a triangle. From groups of order eight it starts to get a little bit complicated. There are five groups of order eight. One is the group of all rotations and reflections of a square. Then there are three groups that are a billion meaning it doesn't matter what order you multiply them all in. Then there's one other group called the quaternion group which is a counter example of almost everything. If you go on and try and classify groups of larger order things start to get extremely complicated. For instance if you go to order 64 there are 267 groups of order 64 that were classified by hand by Marshall Hall. If you go to order two to the ten there are a ridiculously large number of groups which were classified I think it needed some rather hefty computer calculations. However the order of a group doesn't really have much to do with the number of and the size of the order of a group doesn't have much to do with the number of groups of that order. For example there's only one group of order a million and three and that turns out to be because a million and three is a prime number. And how many groups there are of order n doesn't depend so much on the size of n as on the number of primes dividing n. So 1024 is divisible by a really large number of primes all of which two I guess. So there are lots of groups of that order and it seems to be more or less impossible to classify all groups of a given order. If n is a product of a reasonably large number of primes like 10 or 12 or something. For example here's one of the groups of order 64 or at least this is actually a picture of all the subgroups contained in that group. It's got one subgroup of order 64 and some orbits of subgroups of order 32 and then order 16 and order 8 and order 4 and order 2 and order 1. As you can see the structure of this group is really rather complicated. So Hall actually drew diagrams like this for every single one of the 267 groups of order 64. So how can we classify groups well or at least how can we understand groups if we can't classify them? Well the way to do this was done by Galois who sort of depresses everybody because he managed to die at the age of 21 and by that time he had already done work that made himself world famous when most of us are just sort of doing undergraduate exams or something like that. So that was a really impressive, really impressive guy. So what Galois pointed out was that you can break up any group into things called simple groups. The name simple group is a bit misleading they're not really simple at all but a simple group is one that can't be decomposed into smaller groups and a very good analog is to think of groups as being like chemical compounds and you can't possibly classify all chemical compounds they're far too complicated. Here's one particular chemical compounder which you probably recognise as a tiny piece of a molecule of DNA and you just look at this and you would see it's hopeless to classify all compounds. On the other hand every chemical compound is built up of 100 or so atoms and with a certain amount of effort you can classify and find all possible atoms and you get the periodic table. So the relation between groups and simple groups is like the relation between chemical compounds and atoms. Every group can be built out of simple groups in a possibly rather complicated way analogous to the way that every compound can be built out of atoms. In fact, one guy called Ivan Andrus as a sort of joke arranged the simple finite groups in a sort of periodic table. So here is actually a list of, in some sense it's a sort of list of all finite simple groups except it only shows a finite number of simple groups and they're actually an infinite number. By some weird coincidence the finite simple groups consist of 18 infinite families which are arranged here in 18 columns like the periodic table and 26 others and here the analogy breaks down slightly because there should really be 28 rare earth elements. There's no real relation between the chemical periodic table and the finite simple groups. This is just done as a bit of a joke but it's a rather nice looking picture. So what do the simple groups look like? Well there are some simple groups that are easy to describe. These are the cyclic groups of prime order. And you can ask what is the first non-cyclic simple group and this turns out to be the group of symmetries of an icosahedron or at least the group of rotational symmetries of an icosahedron. So here's a collection of objects with the same symmetry group of order 60. So here we have a dodecahedral crystal and here we have a virus. I was trying to find a picture of a coronavirus because that's a rather topical subject at the moment but coronaviruses don't seem to have icosahedral symmetries as far as I can figure out. So here's a different virus. This is a molecule of buckminster fullerene which has 60 carbon atoms and 60 hydrogen atoms and this one down here is a reflection group of order 120 rather than order 60 but it's pretty close to the group of order 60. So if you take a sphere you can reflect in any of these great circles and this gives you a rather nice reflection group. So each reflection turns the black triangles into white triangles and vice versa. Here we have a radiolarian and when I said that Galois was the first to discover the simple group of order 60 that's not really true because they were discovered hundreds of millions of years ago by viruses and radiolarians. Radiolarians are these little animals that form these funny little skeletons. So anyway here's actually if you don't believe viruses have icosahedral symmetry. Here is an actual sort of photograph of an electron micrograph of a virus. It's not entirely obvious that this is an icosahedron so on the right hand side I've marked the vertices with white circles and if you look carefully you can see for instance these three are a sort of triangular face of the virus. Here's a slightly cleaner diagram of it. By the way these circular things making up the virus aren't atoms. The virus is a lot bigger than that. They're actually large-ish protein molecules all fitted together. The virus itself is several hundred atoms in diameter. Anyway here are some more radiolarians so which there was this that they were drawn by this very famous zoologist called Ernst Herkor who actually rather famous for drawing large numbers of these amazingly beautiful slides of various animals. Radiolarians have always kind of puzzled me because there's this question how on earth did they evolve these skeletons. I mean you know in evolution animals and plants evolve by making gradual changes so it's really a bit of a puzzle how you can end up with an icosahedron by making gradual changes. I must admit if someone who believes in intelligent design came up to me and challenged to explain how radiolarians evolved to an icosahedron I'd be kind of stuck. But then I'm not an expert in radiolarians and biologists who study radiolarians actually have considered how they evolved all the articles on this seem to be behind paywalls so I haven't got round to finding out how they actually did it. So how do you classify the finite simple groups? Well the classification of finite simple groups is possibly the most complicated theorem in mathematics that has been written down. Here's a rough picture of the classification. If you look at this pile of books here this pile of books is pretty close to giving you the classification of finite simple groups. So in the middle here we have a series of books by Gorenstein Lyons and Solomon which is where they're carefully writing the classification of finite groups in a form that people can read and mostly the other books are sort of necessary background and some other books covering bits that aren't in this series. Well as you see it's several feet high. The total classification of finite simple groups is estimated to be maybe ten or twenty thousand pages long. It's not actually the longest proof in mathematics but the longer proofs tend to use computers so they will have a computer checking a trillion cases or something like that and obviously that would be longer if you wrote it out but known in their right mind is going to write out a computer calculation that long. So the classification of finite simple groups is probably the longest proof that has been produced almost entirely by humans. So the result of the classification of simple groups as I said is there are 18 infinite families and 26 others and the 26 others are the sporadic groups that I want to say a bit about later on in the talk. So here are some examples of cyclic groups in the icosahedral group we've already mentioned. Some examples of the sporadic groups that I'll be talking a little bit about later are the mature group which is the smallest one with 7,920 elements and the monster group which has this ridiculously large number of elements. So what do the infinite families of simple groups look like? Well we've had one of them the cyclic groups of prime order. The next simplest infinite family of simple groups are the so-called alternating groups. These are subgroups of the symmetric group. So the symmetric group and endpoints is just the group of all symmetries of n objects. You're allowed to permute them and as you know there are n factorial ways of doing that so the symmetric group is order n factorial. The symmetric group isn't quite simple but it's got a simple subgroup with half the number of elements called the alternating group. Another series of simple groups is the general linear group over finite fields. That just means n by n matrices with elements that might be say the integers modulo p and it's not too difficult to work out what the order of this is. And when I said this was a simple group I was actually lying because it's not quite a simple group. In order to make it into a simple group you sort of have to polish it a little bit. First of all you must take the elements of determinant 1 rather than the elements of all non-zero determinant. And secondly you have to kill off the centre to get a group called psl2 of fq. So here p means projective which means quotient out by the centre and s means special which just means take determinant equal to 1. And as well as the general linear groups you can also find analogues of the other groups you get over the real numbers such as what you can define orthogonal groups and symplectic groups and unitary groups over finite fields and get lots of infinite families. So well as I said the proof is 20,000 pages long and here I'm going to summarise the proof in one slide so I'm obviously going to have to miss out a few details. The fundamental key idea of the proof was due to Broward who said you should classify the simple groups by finding centralises of involution. Well an involution means an element of order 2 and the centralise of an involution just means the elements commuting with this element of order 2. So Broward's idea was that you should first find all possible centralises of involution and then for each of those find that simple group with those centralisers. For example suppose you take the centralise of an involution to be a group of order 2 times one of these groups PSL2 that I mentioned earlier and then you get two possibilities. First of all if Q is a power of 3 it gives something called a regroup which is one of the infinite families of groups. One of the particularly difficult infinite families which caused severe problems in its classification. And Yanko caused a sensation in the 1960s by finding that if Q is equal to 5 you actually get a new sporadic simple group. And this was the first sporadic simple group that had been found for almost a century and people got amazingly excited about this at the time and in the next 10 or so years another 20 or so sporadic groups were found. Well in order for this to work you need to show that every finite simple group has an involution which turns out to be equivalent to showing that every finite simple group has even order. And this was proved by Feitem Thompson in the early 1960s and Feitem Thompson's work is more or less a good date for the beginning of the classification of finite simple groups. Their paper is one of the scariest papers I know. Here I've got a copy of it. It's the one volume of the Pacific Journal of Mathematics. Here it's solvability of groups of odd order which is another way of saying that all finite simple groups of even order. And as you see it's a really fat volume of this journal. It's about 250 pages long and the proof is amazingly scary. So here for example I have a definition that goes on for most of this page and quite a lot of this page here too. So they have definitions that are a page long which is more than most proofs you come across. And let's show a random, well it wasn't a random one. I picked a particularly scary page from the proof. So if I focus in on this you can see here they're proving some particular lemma. And they define all these rather complicated expressions and they have this amazingly complicated calculation by generations and relations which is just as scary as it looks. And these sort of calculations go on for the next page and the next page after that. And the next page after that there are more of these long scary calculations and go on for the next page after that. And they finally prove one lemma of these in this several hundred page paper. So I made a couple of attempts to figure out what's going on in this paper and honestly I just have no idea. Normally when I see a mathematical paper I sort of think to myself well I could have proved that if I maybe worked a bit harder or something like that. But this paper by Fyton Thompson it just blows away all my illusions. There's just no way I could ever have proved that theorem no matter how hard I worked on it. I have no idea how Fyton Thompson managed to do that. Anyway, so the first sporadic groups discovered were the mature groups discovered in the 19th century. As I said this was the first one was discovered about a century before Yanko found his. These groups are called M11, M12, M22, M23 and M24, the M stands for mature of course. The numbers 11, 12, 22 and so on are the number of points these groups act on. So M11 for example is a group of permutations on 11 points. In fact it has the property that if you choose any four of these 11 points there's a unique symmetry in the mature group mapping those to any other four points. Which means the order is 11 times 10 times 9 times 8. You can describe the mature group very simply just by saying it's generated by these two permutations. So this is the permutation taking the first object to the second, the second to the third and so on and the 11th to the first. And this permutation takes object 3 to 7, 7 to 11, 11 to 8 and 8 to 3 and then it takes 4 to 10 and so on. So if you just write down these two permutations in one line that defines the mature group. Well obviously writing it down like that doesn't actually give very much insight into the group. You have to do really rather a lot of calculations to find out what M11 actually looks like from those. It's easier to understand a group if you can describe it as symmetries of something. So for M12 you can almost do this as follows. If you take the projected plane of order 3 which is an arrangement of 13 lines and 13 points. Every line has four points on it and every point is on four lines which is why it is order 3 because I guess whoever defined order wasn't very good at counting. But what you can do is you can put 12 counters on the 13 points and you have a sliding block puzzle. You're allowed to move any counter to an empty point on the same line provided you then switch the other two counters on that line. And if you do that you get something that isn't quite M12 it's actually something called M13 and it's not quite a group it's a groupoid. But if you take all the permutations you can do like this that map the empty point to itself that gives you the mature group M12. What I want to talk about next is some groups discovered by John Conway and these groups are groups of symmetries of sphere packings. So the problem is to find the densest sphere packing in any given dimension. One dimension is completely trivial because there's only one way to do it. In two dimensions you can pack spheres on a sort of square lattice like this. But that's obviously not the best way of doing it because you can have a tighter packing by putting them on a hexagonal lattice. In three dimensions this was actually a problem that the Royal Navy was worried about in the 15th and 16th centuries because their ships were carrying cannonballs and didn't have much space and they wanted to know the most efficient way to pack cannonballs. And obviously one way to do it is you can pack them in a sort of pile like this. So how can you pack spheres in three dimensions? Well you can pack them in a sort of cubicle packing like a salt crystal but that's obviously not best. And there are two common ways that crystals can pack spheres. One is called face-centered cubic which looks a bit like this and the other is called hexagonal close packing. And the difference between face-centered cubic and hexagonal close packing can be described as follows. Suppose you start by taking all these white spheres and packing them in a plane. Then you want to put some black spheres in the next layer above them by putting them on the gaps between the white spheres. So you get these black spheres here. Then on the third layer you want to put some more white spheres but it turns out there are two ways of doing this because you can either put the white spheres on the third layer above the white spheres on the second layer or you can put them not above the white spheres. So you can see this sphere C is actually above one of the gaps in the first layer. And in each layer there are up to three possible configurations you can put the spheres in which are normally labelled as A, B and C. So the only constraint is that the spheres on one layer can't be in the same positions as the spheres on the layer just below it. So you can describe a packing of spheres by having a sequence of letters each which is A, B or C with no two adjacent letters being the same. For instance, hexagonal close packing you just do A, B, A, B, A, B and so on. For face-centered cubic you do A, B, C, A, B, C, A, B, C and so on. But there are an uncountable number of other equally dense packings. So you can ask what's the best sphere packing in any given dimension. So dimension one is trivial, dimension two is easy. Dimension three, face-centered cubic is the densest possible packing. But as I just said, it's not unique. Four to seven, we don't know the answer. So you can ask what's the best sphere packing in any given dimension. Four to seven, we don't know the answer. Well, we sort of have a pretty good idea what the answer is but we haven't managed to prove it. Dimension eight, the densest lattice packing is something called the E8 lattice that I'll describe a bit later. Dimension nine to 23 again, we have a guess about what the best answer is but we haven't managed to prove it. And in dimension 24, the densest packing is called the leach lattice. And in this case, it is unique. Dimensions greater than 24 again, we don't know. And this is very weird because you would expect dimension four to be the next easiest case after three. But it turns out the next easiest case is after three at eight and 24. So proving the densest sphere packing in any given dimension is often an extremely difficult problem. So in three dimensions, this was solved a few years by by Hales. Hales' proof is incredibly difficult. It's not only takes up several hundred journal pages, it also uses several terabytes of computer calculations. So it's long both in terms of journal pages and in terms of computer calculation. The proof in eight dimensions is very much shorter. This was found by Marina Viersowska quite recently. She was extending work done by Henry Cohn and Nirm Elkes who showed that E8 would be the densest eight-dimensional sphere packing provided you could find a function with several rather extraordinary properties. And quite a few people tried to find such a function. I mean, I spent a few hours on it and just made no progress at all and couldn't figure out how to do it. Marina Viersowska came up with this incredibly ingenious and amazing construction of this function to show that E8 is the densest lattice packing in eight dimensions. And a minor variation of her construction also shows that the leach lattice is the densest in 24 dimensions. So far, no one has managed to find similar magical functions in any other dimension. So how do you actually describe these sphere packings? Well, one way to describe a sphere packing is to give the coordinates of the centers of the spheres in the packing. For example, to describe the cubicle packing, you'll just say that the centers of the spheres are all coordinates A, B, C, where A, B and C are arbitrary integers. And you can see immediately there are six spheres touching a given sphere and the spheres touching a given sphere have these coordinates. Face centered tube because kind of similar except you apply the restriction that some of these integers has to be even. In this case, you can easily see there are 12 spheres that are closest to the origin given by these 12 points. So we can now describe the E8 packing of spheres. It's almost where you take all numbers to integer coordinates, except that first of all, you say that the sum is even as in the face centered cubic. But then you also allow all the coordinates, they don't all have to be integers, they could all be integers plus a half. And now if you try and find the spheres closest to the origin, it turns out there are two ways of writing them. First of all, all coordinates could be a half with some signs and there are 128 of these. You might think there are 256 because there's a two choices of each sign, but the sum has to be even and that cuts down the number by a factor of two. Secondly, you could have two coordinates that are plus or minus one. And the number of these is 8 times 7 over 2 for the two choices of coordinate times 2 squared for the possible sign. And if you add these up, you discover there are 240 spheres touching a given sphere. And if you want to see what these 240 spheres in eight dimensions touching a given sphere look like, well, you can't really draw them in eight dimensions, but if you project them down to two dimensions, it ends up looking something like this where each of these points is the projection of one of these 240 spheres into two dimensions. And you draw lines between two points depending on the distance of the two spheres. So you can ask what is the symmetry group of this configuration of 240 spheres? And it turns out to be a reflection group. So here are a few examples of reflection groups or at least things whose symmetry group is a reflection group. So the first three are sort of Euclidean reflection groups. You can see the symmetries of each of these patterns are generated by reflections. For instance, in this third one you can reflect in a line like that which will preserve the pattern only changing the blue to green and so on. The bottom three are examples of spherical reflection groups. Here what you do is you take a sphere and you reflect it in one of the hyperplanes of these three diagrams and that gives you a spherical reflection group. The groups of orders 24, 48 or 120 and turn out to be the full automorphism groups of a tetrahedron or octahedron or icosahedron including reflections. Finally these two diagrams are examples of hyperbolic reflection groups. These are models of the hyperbolic plane. So the triangles in this picture look as if they've got curved sides and they look as if they're all different sizes but really all these triangles are straight sides and they're all the same size because when you draw a hyperbolic plane you have to distort it a bit to make it fit into Euclidean space. So these give two examples of reflection groups that live in hyperbolic space. So the symmetry group of the E8 lattice is the E8 reflection group and you can think of it as being a bit like the symmetries of a cube only in very much higher dimensions. So a cube has, you can reflect in the hyperplanes orthogonal to vectors that are either that form or this form and this gives you the set of generators for the reflection group of a cube and the E8 reflection group instead of having nine reflections the cube has has 120 reflections corresponding to half of the 240 vectors I wrote down earlier. So that sort of gives you a picture of the symmetry group of the E8 lattice it's generated by these 120 reflections that you can write down fairly explicitly. Next we move on to the sporadic group discovered by John Conway and this is a picture of John Conway exhibiting one of his talents that he was particularly proud of which is that he could make more shapes with his tongue than than anybody else in the world. Here he's demonstrating one particularly intricate one. Anyway so John Conway discovered the Conway sporadic simple groups by studying symmetries of sphere packing so we've seen three of these already so in two dimensions the hexagonal sphere packing has a symmetry group of order 12 in three dimensions the face centered cubic has a symmetry group of order 48 we just looked in eight dimensions where the E8 lattice has a symmetry group of this order which isn't too hard to work out and has 240 spheres touching one sphere Conway, John Conway studied the leached lattice where each sphere touches 1 9 6 5 6 0 other spheres and the symmetry group has this massive order which is more or less one of the Conway groups it's actually twice the size of Conway's simple group but never mind. So when Conway discovered this he said he you know he was he was basically jet setting all over the world giving lectures on this group and what was really exciting group theorist was that he not only discovered three different groups as symmetry groups of configurations in this lattice but he also found that most of the other sporadic simple groups that had been discovered before then were also fitted inside his group and so all the mature groups for example live inside the Conway group the Yanko group doesn't quite fit inside the Conway group so he didn't quite explain all the known sporadic simple groups so next I want to show you how you construct the leached lattice and well let's go back to cannonballs in a pyramidal pile and ask how many cannonballs are there well it's pretty easy you just take one squared for the first row two squared for the second row three squared for the third row and so on so the total number of cannonballs is the sum of the first n squares and now we have the following recreational mathematical problem suppose the number of cannonballs in this pile is a square and there's more than one cannonball how many cannonballs are there and the answer turns out to be there have to be 4,900 because that's the only way the sum of the first n squares can itself be a square and it turns out you can use this funny fact to construct the leached lattice so what we do is we take 26 dimensional space with a funny distance in it I'm not going to use usually Euclidean distance I'm going to use a Lorenzian distance so if you've done special relativity you've come across this funny Lorenzian metric where the distance between a point from the origin is given by the square root of the sum of the squares so let's put a minus sign in front of one of the coordinates and now I'm going to take the following vector w with coordinates 70, 0, 1, 2, 3 and up to 24 and now w has length zero so if you're doing general special relativity you would call w a light like vector and the leached lattice can now be constructed from vectors in this 26 dimensional space so first of all all coordinates are either integers or are all integers plus a half and the sum of the coordinates is even so these two conditions are rather like what we had for the E8 lattice and the third condition is that the vector has to be orthogonal to the vector w, meaning it's Lorenzian in a product with w is zero and strictly speaking we should also have quotient out by w but I won't worry about that too much so Conway's group is more or less the group of automorphisms of this funny lattice defined by the first two conditions that also fix this vector w this weird construction only works in 24 dimensions there doesn't seem to be any good analog of it in any other dimension there's something very odd about 24 so that more or less describes Conway's simple group and now in the rest of the talk I'll be trying to explain John Mackay's t-shirt so John Mackay had a specially made t-shirt that he used to wear to mathematical conferences explaining his great discovery that 196883 plus 1 is equal to 196884 and the significance of this is that 196883 turns out to be the dimension of the space in which the so-called monster sporadic simple group lives and 196884 turns out to be one of the coefficients of the so-called elliptic modular function and for some time when John Mackay first pointed this out several people said it was just this meaningless coincidence that if you get lots of numbers then every now and then two of them will be really close just by coincidence for example e to the pi is very close to being pi plus 20 as explained in this XKCD comic and maybe 196883 and 196884 are close just as a random coincidence well here are some other random coincidences if you take e to the pi root n it's quite often very close to an integer if you take e to the pi as in the previous slide it's not very close to an integer but it's very close to an integer plus pi although that's probably just a meaningless coincidence and the explanation of these turns out to be also related to the elliptic modular function so the elliptic modular function was introduced by Felix Klein here's a picture of him with a every German mathematician in the 19th century had this formidable beard so here's Klein with his mathematical beard he's best known for inventing the Klein bottle which is this funny one-sided surface here's a sort of picture of it incidentally it was originally called a Klein surface not a Klein bottle and it ended up being called a bottle because somebody mistranslated the German you see the German words for surface and the German words for bottle were actually very similar and easy to confuse anyway here is Klein's elliptic modular function and the definition looks like a silly joke at first you write down this random bizarre expression and you can work out its power series expansion and it turns out to have these rather complicated coefficients so where on earth does this function come from? well it turns out to be the simplest possible function that is invariant by adding 1 to tau and also invariant under changing tau to minus 1 over tau so adding 1 to tau is not a big deal any function of e to the 2 pi i tau will be invariant like that but making it also invariant under tau goes to minus 1 over tau is very much more difficult and this really is the simplest way of doing that so here's a picture of Klein's elliptic modular function these are some absolutely stunning hand-drawn pictures in this book by Janker and Emder and what's going on here is this is the upper half plane in the complex numbers and this is actually a graph of the absolute value of Klein's elliptic modular function it's considered as a function from complex numbers to complex numbers and you see it has this series of poles it's pretty obvious why poles are called poles if you look at this diagram and they haven't actually drawn all these poles because there really should be an infinite number of poles all along the real axis which is this line here they've only drawn 5 representative poles and there should really be more and more and more of them getting smaller and smaller but I guess their patience ran out so strictly speaking this isn't actually Klein's modular function but a slightly different modular function anyway so Klein's modular function explains why e to the pi root 163 is an integer because the elliptic modular function is the weird property that it's an algebraic number whenever tau is an imaginary quadratic irrational and it just happens to be an integer when tau is equal to i tau is equal to minus 1 plus i root 163 over 2 so the result of this is that if q is equal to minus e to the pi root 163 if q is equal to that then this expression here is an integer and this number here is equal to e to the pi root 163 and this is an integer and this term here is incredibly small I mean it looks big because of the 1, 9, 6, 8, 8, 4 but q is so tiny that all these terms here are incredibly small and the result of this is that e to the pi root 163 is very close to this integer anyway, I now want to explain how Klein's modular function is related to the monster group so the monster group is the largest of the spadic groups and it was discovered by Fisher and Grice in the early 1970s and the smallest simple group is 60 symmetries and lives in three dimensions Conway's group which was thought to be huge when it was discovered lives in 24 dimensions and has this large number of symmetries the monster group lives in 1, 9, 6, 8, 8, 3 dimensions and has this absurdly large number of symmetries this number was first calculated by Robert Grice so you might ask well how on earth do people even come across such a group well I think they came across it by looking at centralises of an involution that Fisher had discovered a rather smaller group now called the baby monster and a slight variation of it is the centre of an involution in the monster group so you find it by using Brower's idea that you should study groups by looking at their centralises of involution and it was first constructed by Robert Grice which absolutely stunned everybody I mean I was sort of in Cambridge with John Conway when Robert Grice came out with this construction John Conway told me that everyone doing groups here at Cambridge was completely flabbergasted by his construction I mean they're all known in principle that you could construct the monster group by defining an algebra structure in 1, 9, 6, 8, 8, 3 dimensions but everyone assumed that this calculation would be so incredibly complicated that it would be impossible for anybody to carry out in practice but Robert Grice actually managed to do this incredible calculation he apparently spent several months working non-stop every day in the Princeton library and finally managed to define an algebra structure on this space and show that its automorphism group was the monster so here's a piece of the monster character table what the character table is is it tells you what dimensions the monster lives in so this first column you see this entry 1, 9, 6, 8, 8, 3 tells you the monster lives in 1, 9, 6, 8, 8, 3 dimensions the next entry tells you the monster is also the symmetries of something in this number of dimensions and so on this is only a tiny piece of the character table of the monster the full character table has 194 rows and 194 columns and if I displayed that you wouldn't be able to read any of it so I've just given you the top left hand corner of it so if you look at this number 1, 9, 6, 8, 8, 3 was one of the numbers appearing on John Mackay's t-shirt and monstrous moonshine as discovered by Mackay Thompson, Conway Norton one aspect of it says that each of the coefficients of the elliptic modular function except for this coefficient here is a simple combination of dimensions of representations of the monster so here I've written out the first three or four of them now the equality of 1, 9, 6, 8, 8, 4 and 1, 9, 6, 8, 8, 3 plus 1 might just be a coincidence but when you see the next few cases it's obviously not a coincidence I mean you might get one of these relations holding my accident but getting three or four of them indicates there's something funny going on 744 doesn't appear but 744 is just a historical accident the point is the elliptic modular function is really only defined up to a constant and the constant happens to be 744 for historical reasons, not for any good mathematical reasons so you can sort of set it to 0 if you want so John Thompson suggested that the reason for this might be that the monster acts on a graded vector space of infinite dimension which is the direct sum of pieces of finite dimension and the pieces of finite dimension have dimension equal to a coefficient of the elliptic modular function and such a vector space was constructed by Frankl Lepowski and Merman in possibly in the early 1980s their construction actually used ideas from string theory so string theory is this possible mathematical theory of everything physicists have been trying to make it tie up with actual physics for several decades and it's still not clear exactly what its status is in the early days of string theory it only worked really well in 26 dimensions which was a bit of a problem for physicists because space-time is 4-dimensional not 26-dimensional however 26 dimensions may have been a problem for physicists but it actually was really nice for mathematicians because it turns out to be exactly what is needed for constructing the monster and the leech lattice you remember I constructed the leech lattice from a 26-dimensional Lorentzian space well it's the same 26-dimensional Lorentzian space that turns up in string theory and the construction of the monster vertex algebra turns out to be rather closely related to string theory in 26 dimensions in particular Frankl Lepowski and Merman used so-called vertex operators to construct it where vertex operators are certain operators that appear in string theory they're sort of related to the name vertex operator comes from the fact they sort of destroy what happens with the vertex of a Feynman diagram this monster space also turns out to have a rather complicated algebraic structure called a vertex algebra structure that's acted on by the monster so even if physicists decide they're not interested in string theory after all and it doesn't work, mathematicians will still be interested in the string theory because there's a lot of rather nice mathematics going on in there actually the relation to moonshine is only a minor mathematical aspect of string theory it also turns up in all sorts of other things like mirror symmetry in the CFT-ADS correspondence and things like that well these rather bizarre relations between the monster-spratic group and elliptic modular functions also hold for many of the other spratic groups for example let's take the baby monster group that the monster was constructed from then it lives in spaces of dimension 4371, 96256 and so on and there's another elliptic modular function whose coefficients look like this and again you can see 4372 is pretty darn close to 4371 and so on so everything you can do for the monster-simple group has a sort of analog for the baby monster and in fact most of the spratic groups have similar relations to modular functions they have representations whose dimensions are given by some sort of modular function well I'll finish off by just by describing some open problems first of all there's the problem of umbral or nemire moonshine so Ramanujan just before he died sent a letter to Hardy where he described some weird functions he found called mock-theta functions and here's an example of a mock-theta function it has this strange-looking power series expansion on the other hand the mature group M24 that I mentioned earlier has representations in the following dimensions 2345231 and so on now if you look at the numbers 45231, 770, 2277 and so on they are the coefficients of this mock-theta function so I think this was originally observed by Eguchi, Guri and Tashikawa and Chen and Duncan and Harvey then pushed this a lot further and showed there were lots of bizarre relations between the group M24 and M12 and other groups and certain nemire lattices which are rather similar to the leach lattice there are certain lattices in 24 dimensions so as far as I know people have verified that the coefficients of these mock-theta functions are indeed closely related to representations of these groups but nobody knows why for example you could ask is there some sort of algebraic structure underlying this similar to vertex algebras as far as I know nobody has yet found one there's a little bit of a problem there's this coefficient minus 1 here so it looks as if you might need a minus 1 dimensional vector space whatever that means which could be a little bit tricky to find there are also some philosophical open problems about sporadic groups that have bugged me and many other people if you look at the actions for a group they're incredibly easy and natural you know they just sort of take half a line to write out but somehow inside these axioms are hidden things like the monster group and the sporadic group which are mind-bogglingly complicated so it just seems very strange that these very simple axioms should be hiding such complexity in them so many people have wondered is there some simpler or more uniform construction of all the finite simple groups or is there some much simpler explanation of the classification of finite simple groups I mean could we reduce the 20,000 pages of the classification to 50 or 100 pages that humans might actually be able to understand you know maybe not maybe people have been trying to simplify the classification for several decades and they've made some smallish improvements I mean you can maybe reduce it from 20,000 pages to 10,000 pages or something but maybe it really just is that complicated I'll just finish by listing some places to find out more there's a popular book on the monster of moonshine by Ronan if you're interested in a more advanced introduction to this the book by Conway and Sloan on Spherepacking's lattices and groups describes the construction of the leech lattice and M24 and the monster and most of the other groups I've been mentioning if you'd like a general survey of the classification of finite simple groups there's a really nice book called Finite Simple Groups by Gorenstein that describes this