Here's an attempt at a tl;dr. Disclaimer: I am not an expert in this stuff. Further disclaimer: it's all a bit technical, and while I can explain what these peo

Here's an attempt at a tl;dr. Disclaimer: I am not an expert in this stuff. Further disclaimer: it's all a bit technical, and while I can explain what these people proved I can't tell you why it's interesting, and without that motivation it may be heavy going. I'll try to make it accessible-in-principle to readers with little mathematical knowledge, though whether they'll be interested is anyone's guess. Further further disclaimer: this is long enough that I need to split it into two comments. Sorry about that. This is comment number 1. OK. So, ingredient number 1 is a thing called SL(n,Z). To approach this, begin by imagining a square grid, extending infinitely in all directions. Mathematicians call this Z^2, "Z squared", where for historical reasons Z means the integers -- the whole numbers, positive, negative or zero. Now, there are a number of geometric transformations you can do to this grid. We're interested in what are called linear transformations from Z^2 to itself, which means that if you have (x,y) in the grid then the point (u,v) that it maps to is always also in the grid, and that if (x,y) maps to (u,v) and (x',y') maps to (u',v') then (x+x',y+y') always maps to (u+u',v+v'). (Note: the usual definition of "linear" is a bit more complicated, but in this particular case that's enough.) A nice simple example of such a linear map is rotation by 90 degrees (anticlockwise, let's say) about the origin. So e.g. (2,3) maps to (-3,2). A less obvious one maps (x,y) to (7x+5y,-4x+3y). In general, any linear transformation from Z^2 to Z^2 maps (x,y) to (ax+by,cx+dy) for some integers a,b,c,d. Now SL(2,Z) consists of just some of these maps. The condition for being in SL(2,Z) is a little peculiar: it's that ad-bc=1. Why that? Well, this says two things. The first, and more important, is that the map is invertible: there is another linear transformation from Z^2 to itself that undoes what it does. (The map has an inverse that might not send Z^2 to itself provided ad-bc is nonzero; its inverse sends Z^2 to itself provided ad-bc is either +1 or -1. If you happen to have recognized the determinant of a 2x2 matrix then you are thinking on the right lines.) The second condition, which picks out the transformations with ad-bc=+1 rather than =-1, is that the maps should be "orientation-preserving". One way to say this: take a triangle, defined by three points in Z^2, and hit it with the transformation. Suppose you go around those three points in anticlockwise order; if the transformed points are then also traversed anticlockwise rather than clockwise, the transformation is orientation-preserving rather than its opposite, "orientation-reversing". Rotations and "shearings" are orientation-preserving; reflections are orientation-reversing; you can build any of these transformations from those ingredients; if you can do it without using reflections then your transformation is orientation-preserving. OK. We have constructed this thing called SL(2,Z). It turns out to be surprisingly important in various areas of mathematics. Zimmer's conjecture is about something one step more generalized: SL(n,Z). To make this, start with the n-dimensional version of that square lattice: each point in it is just an n-tuple of integers. This is called Z^n. Now, once again, consider the linear transformations from Z^n to itself (same definition as above); once again, you can define such a transformation by a bunch of integers; once again, there is a number called the determinant that tells you whether the transformation is invertible and orientation-preserving; once again, take the transforations with determinant 1. These form SL(n,Z). This object has some important structure to it. One aspect of this is that it's what mathematicians call a group. Given any two elements of SL(n,Z) -- any two of these transformations -- you can do one and then the other, getting what's called their composition. The composition of two things in SL(n,Z) is always itself in SL(n,Z). And the inverse of anything in SL(n,Z) -- the thing that undoes what it does -- is always in SL(n,Z). A set of transformations with these properties is called a group (note: this is not quite right, and there are also groups that aren't groups of transformations, and the official definition looks more technical), and it turns out that groups are important all over mathematics. [End of comment number 1. Continues in comment number 2.]