 This talk will be about the leach lattice, or more precisely about a very short construction of the leach lattice found by John Conway and Neil Sloan. So the first problem is what is the leach lattice and for that matter what is a lattice? Well a lattice we recall just means a regular arrangement of points. So for example in two dimensions we could have a square lattice, like the points on the vertices of a chessboard and go on forever, another lattice in two dimensions would be a triangular lattice where you arrange them at the vertices of an equilateral triangle and there are lots of other two-dimensional lattices. A good way of thinking of a lattice is you can think of it as being a packing of spheres. So you can think of these points as being the centers of spheres, so here the spheres would look like this. I'm calling them spheres you may say they're circles not spheres but a circle is a sphere in two dimensions and more generally I want to talk about lattices in various other dimensions and here the spheres or possibly circles will kind of look like this and so on. So these are packings in two dimensions in three dimensions you could think of a lattice as being something like the centers of the atoms in a crystal and the spheres might be the sort of solid atoms touching each other. It turns out that lattices are very special in 8 and 24 dimensions which is very weird in fact this wasn't really noticed until relatively recently I mean people spent 100 or 200 years studying lattices without noticing that these two dimensions are very special. Why are these dimensions so special? I don't know and I don't think anybody else does either maybe it's you know sort of a practical joke played on us by the universe. In particular in 8 and 24 dimensions there are two very special lattices. In eight dimensions it's called E8 for complicated historical reasons and in 24 dimensions it's called the leach lattice because it was discovered by John Leach in the early 1960s. It may have been discovered by Witt about 25 years earlier but exactly what Witt discovered is a little bit murky. The leach lattice turns out to be related to loads of other things it became really famous when John Conway used it to construct various simple groups by looking at the symmetries of the leach lattice. It also turns out to be the densest possible packing of spheres in 24 dimensions. Marina Via Zovska found an incredible method for proving that certain lattices are the densest possible ways to pack spheres and what is really weird about her method is that it only works in 8 and 24 dimensions. It's really absolutely bizarre. The leach lattice also gives you things like what is the maximum number of spheres that can touch a given sphere. For instance in two dimensions it's pretty obvious that you can have six spheres touching a sphere and no more. In 24 dimensions the maximum number is given by the leach lattice and is 1 9 6 5 6 0. So how do you construct the leach lattice? Well the original construction by John Leach was very complicated and involves things like Golay codes and Steiner systems which are very interesting but the only problem with them is they're a little bit complicated to define and describe. And so instead what I'm going to do is I'm going to give an astonishingly short and simple construction of the leach lattice found by Conway and Sloane which uses properties of Lorentzian space. So you remember in ordinary Euclidean space we define the distance by taking the sum of squares of coordinates and taking the square root of that so that gives you the distance of a point from the origin. If you do special relativity then you use a special Minkowski metric where you take x squared plus y squared plus z squared minus t squared and you notice that there's this funny minus sign here which makes everything very strange. So in particular if you try and draw a picture of Lorentzian space well I'm going to sort of draw three-dimensional space. You think of a three-dimensional space here there's an x axis and a y axis and a z axis. If you take all the vectors of norm zero this forms something called the light cone so that tells you how light travels and the vectors outside here are called space-like and the vectors inside these cones are called the timeline vectors. In Lorentzian space you can still form lattices for instance you can take the lattice of all points x, y, z and so on up to t with all x, y, z and t in the integers say so that would give you a sort of analog of the square lattice. I'm going to look at Lorentzian lattices in 26 dimensions. So this is called x naught x1 up to x24 and then we've got a timeline vector t and the distance is going to be taking the square root of x naught squared plus x1 squared plus x24 squared minus t squared. Conway and Sloan took the following magical vector 0, 1, 2, 3 up to 24, 70 and this is called a vial vector drawn with a Greek letter rho. This vector is the following funny property the inner product of rho with itself is equal to zero squared plus one squared and so on plus 24 squared minus 70 squared and by some freaky coincidence this is equal to zero and 24 is the only number for which this holds apart from trivial cases like naught squared equals naught squared and naught squared plus one squared equals one squared so apart from these two trivial cases 24 is the only number for which the sum of the first n squares is a square. Now with this vial vector we can now form a lattice as follows. We take the lattice of all points with all coordinates integers which is called i25 1 and inside this we take your throttle complement of the vector rho so this now has dimension 25 and now because rho has inner product zero with itself rho is actually in the 25-dimensional lattice that's orthogonal to rho. If you were used to working in Euclidean space this sounds kind of crazy that if you take the orthogonal complement of a vector the vector is not in that but it can happen in Lorentzian space so we can take this 25-dimensional lattice and just kill four copies of rho and this now has dimension 24 so we've got a 24-dimensional lattice and this lattice here is not the leach lattice however it's very very close to being the leach lattice just take all the vectors x0 x1 x24 t with the sum even here remember these are all integers and i'm going to insist that the sum must be even so that gets rid of half of them because the sum is either even or odd and now we add in all the translates to half a half and make all its coordinates a half so we sort of throw away half with then we double it up and if you do this then rho perp over rho is now the leach lattice this is particularly bizarre because the automorphism group of the leach lattice is one of conway's sporadic simple groups and normally sporadic simple groups are incredibly difficult to define explicitly this is by far the simplest way of explicitly constructing a sporadic simple group as i know of incidentally this trick of throwing away half the lattice and adding half of something back else can also be used to construct the e8 lattice so if we take the sum of eight copies of z and throw away half of it and add in the translates by this as before we just get the e8 lattice in eight dimensions okay if you want to know more about this the best place to find out is the book sphere packings lattices and groups by conway and slone and you can find their construction of the leach lattice in chapter 26 let me just magnify it a bit for you so you can read it so here on page 526 we find this vector with coordinate 0 up to 24 followed by 70 which they use to construct the leach lattice