More sketches, and why we care
Dr. Adams just sent me a link to an explanation of the technical details for mathematicians in other fields, but it’s still somewhat readable.
I also have been reading the slides for Dr. Vogan’s talk, The Character Table for E8, or How We Wrote Down a 453,060 x 453,060 Matrix and Found Happiness. There’s also an audio recording available (7MB mp3). Incidentally, I’d have gone for The Split Real Form of E8, or How We Learned to Stop Worrying and Love the Character Table, but it’s all good. This talk actually manages to be very generally accessible, and includes all sorts of pretty pictures. Those of you who wanted more visuals than I provided in my rough overview might like to check that one out.
Together, these two are my core that, together with some input from Dr. Zuckerman I’ll be trying to break down into smaller chunks. I highly advise reading at least Vogan’s slides and preferably also Adams’ notes.
I also want to respond to a comment basically asking, “so why the heck should we care about this?” It’s an excellent question, and yet another one the newspaper reports really glossed over without taking seriously. I’ll admit that I glossed it over at first too, since I think this stuff is just too elegant not to love. Still, I’ve mulled this over not just as applies to these calculations, but with regard to a lot of mathematics at this level (thus qualifying the “why we care” as a rant).
This sort of question from a non-mathematician almost always is looking for an engineering response. “What’s it good for?” means, “what can we build with it?” Honestly I have to say “not much”. Representations of Lie groups do have their uses, though, and I can point out a few things they have already been good for.
As indicated in Dr. Vogan’s slides, representations of the one-dimensional Lie groups are concerned with change through time, particularly periodic changes. This means that they’re exceptionally good at talking about periodic phenomena, like waves. Sound waves, light waves, electrical circuits, vibrating strings — they’re all one-dimensional waves. So what? So every time you use the graphic equalizer on your stereo the electronics are taking the signal and performing a fast Fourier transform on it. This turns a function on the line (Lie group) into a function on the space of all representations of the group; that’s the “unitary dual” that Dr. Adams refers to. Then you can adjust the periodic components and reconstruct a new function with much fatter bass, or whatever your tastes are.
The same sorts of things can be done in higher dimensions. Similar techniques revealed that you can’t hear the shape of a drum — there are differently-shaped membranes that have the same vibrational characteristics. What are “orbitals” of electrons around an atomic nucleus (hazy memories of chemistry)? They’re representations of the Lie group !
So what can we do with ? Nothing right now, but there’s plenty we can do (and have done) with representation theory in general.
There’s another reason (beyond the intrinsic beauty of the ideas) to work out the Atlas: more data means more patterns, and more patterns means more interrelationships between seemingly-distinct fields. Quite a few of the greatest theorems in recent years have been saying that this field of mathematics over here and that one over there are “really” the same thing. Everyone knows that Andrew Wiles solved Fermat’s Last Theorem, but what he really did was show that some things in algebraic geometry (the study of solution sets of polynomials) called “elliptic curves” are deeply related to functions with a certain sort of periodicity called “modular forms”. If, as David Corfield asserts, mathematics proceeds by “telling stories”, then each field’s stories become allegories for the other. Hard questions in one area might be translated into questions we know how to solve in the other.
So how does having a lot of data like the Atlas around help out? Because we discover a lot of these relationships from similar patterns in the data, and in many cases (though I hate to admit it) through the same numbers showing up over and over. As just one example, I present the Monstrous Moonshine conjecture. The Monster is a finite, simple group — no normal subgroups, so it can’t be broken down into even a semidirect product of smaller groups — of order (brace yourself)
That’s elements being juggled around in an intricate symmetry. People sat down and calculated its character table, very much a similar project to the current one about . And then there’s a certain special modular form called that just happens to be related to it. How so? John McKay happened to see the -function written out like this:
So? So he’d also seen the dimensions of representations of the Monster, which start with , , , and continue. Every single coefficient in the function came from dimensions of representations of the Monster! And it was conjectured that the pattern continued. In fact it did. Twenty-some years ago, Frenkel, Lepowsky, and Meurman constructed a representation of the Monster that made it clear, and their results are still echoing. One of my colleagues graduated last year and went on to Harvard by studying exactly the same sorts of connections.
And how did it start? By recognizing patterns in a mountain of raw data about representations. What unsolved problems might be translatable into representation theory by reflections found through the Atlas data? Maybe the Navier-Stokes equations, which would give a better understanding of fluid flows and aerodynamics. Maybe the Riemann hypothesis, which would lead to a better understanding of the distributions of prime numbers, which would have an impact on modern cryptography. Who knows?
Oh, and one more thing. How did someone find the Monster in the first place? Well it turns out to be a group of symmetries of a certain collection of points tiling eight-dimensional space. What collection of points? The “Leech lattice”. And you’ve already seen it: that picture of the root system in all the news reports is the basic cell, just like a square is the basic cell of a checkerboard tiling of the plane. And it all comes back around again.
How the heck can you not care about this stuff?
[EDIT: I’ve found out I was wrong about how the Monster relates to . More info in the link.]