A rough overview
I’ve had a flood of incoming people in the past couple days, and have even been linked from the article in The New York Times (or at least in their list of blogs commenting on the news). As I said before, their coverage is pretty superficial, and I’ve counted half a dozen errors in their picture captions alone.
One of the main reasons I write this weblog is because I believe anyone can follow the basic ideas of even the most bleeding-edge mathematics. Few mathematicians write towards the generally interested lay audience (“GILA”) the way physicists tend to do, and when mathematics does make it into the popular press the journalists don’t even make the effort they do in physics to get what they do say right.
My uncle, no mathematician he but definitely a GILA member, emailed me to mention he’d read that mathematicians had “solved E8″, but had no idea what it meant. Mostly he was asking if I knew Adams (I do), but I responded with a high-level overview of what they were doing and why. I’m going to post here what I told him. It’s designed to be pretty self-contained, and has been refined from a few days of explaining the ideas to other nonmathematicians.
Oh, and I’m not above link-baiting. If you find this coherent and illuminating, please pass the link to this post around. If there’s something that I’ve horribly screwed up in here, please let me know and I’ll try to smooth it over while keeping it accessible. I’m also trying to explain the ideas at a somewhat higher level (though not in full technicality) within the category “Atlas of Lie Groups”. If you want to know more, please keep watching there.
[UPDATE: I now also have another post trying to answer the "what's it good for?" question. That response starts at the fourth paragraph: "I also want to...".]
I understand not knowing what the news reports mean, because most of them are pretty horrible. It’s possible to give a stripped-down explanation, but the popular press doesn’t seem to want to bother.
A group is a collection of symmetries. A nice one is all the transformations of a square. You can flip it over left-to-right, flip it up-to-down, or rotate it by quarter turns. This group isn’t “simple” because there are smaller groups sitting inside it [yes, it's a bit more than that as readers here should know. --ed] — you could forget the flips and just consider the group of rotations. All groups can be built up from simple groups that have no smaller ones sitting inside them, so those are the ones we really want to understand. Think of it sort of like breaking a number into its prime factors.
The kinds of groups this project is concerned with are called Lie groups (pronounced “lee”) after the Norwegian mathematician Sophus Lie. They’re made up of continuous transformations like rotations of an object in 3-dimensional space. Again, the Lie groups we’re really interested in are the simple ones that can’t be broken down into smaller ones.
A hundred years ago, Élie Cartan and others came up with a classification of all these simple Lie groups. There are four infinite families like rotations in spaces of various dimensions or square matrices of various sizes with determinant 1 (if you remember any matrix algebra). These are called , , , and . There are also five extras that don’t fit into those four families, called , , , , and . That last one is the biggest. It takes three numbers to describe a rotation in 3-D space, but 248 numbers to describe an element of .
Classifying the groups is all well and good, but they’re still hard to work with. We want to know how these groups can act as symmetries of various objects. In particular, we want to find ways of assigning a matrix to each element of a group so that if you take two transformations in the group and do them one after the other, the matrix corresponding to that combination is the product of the matrices corresponding to the two transformations. We call this a “matrix representation” of the group. Again, some representations can be broken into simpler pieces, and we’re concerned with the simple ones that can’t be broken down anymore.
What the Atlas project is trying to do is build up a classification of all the simple representations of all the simple Lie groups, and the hardest chunk is , which has now been solved.