I’m about to head off to participate on the “alumni” team in a scrimmage for the Howard County and Baltimore County teams going to the American Regions Math League.
As the term “alumnus” connotes, I did this stuff myself back in high school. To be honest, I thought that it was pretty silly even then. It ends up emphasizing speed and trivia over deep understanding of mathematics. The various Olympiads are better, but still not great. There’s something in the society at large, though, that wants to reduce every single human activity to a contest, and mathematics for high school students is no exception. If I hadn’t already been studying more advanced material on my own, I could easily see ARML beating the enjoyment of mathematics out of me.
Still, some kids like running the races and like memorizing a billion little factoids. If they enjoy it, fine, and it’s close enough to real mathematics to make it worth encouraging. And so I do my part.
I’m wrapping up my coverage of ring theory (for now). There’s a lot I’ve left unsaid about rings, and also about groups. I’m hoping, though, that I’ve given a certain amount of a feel for how algebraic structures work in preparation for the next topic: categories.
There are a number of readers, I know, who have been waiting for this point almost as much as I have been. There are also some who are dreading it. Everything up until this point has been stuff that everyone has to know, but categories are still a bit controversial in some circles. Many people find them even more abstract, or technical, or even content-free than other parts of algebra.
Category theory is at turns praised and derided with the same phrase, “abstract nonsense”. Indeed the earliest uses were to make general statements about algebra, just like ring theory makes general statements about polynomials, and polynomials make general statements about numbers. For some reason there are still mathematicians who draw a line in the sand and say, “Here! No further!”, just as others saw it as the next natural step.
Personally, I have been drawn to categories since I knew they existed. I still remember being shown the natural transformation from the identity functor on the category of vector spaces over a given field to the double-dual functor, and going back to Jeff Adams’ office (yes, the same Jeff Adams) again and again for more back in the spring of 1999. I hope now to say what it is that I saw then (and still see) in category theory, and to make the case for them. I really, honestly believe that within the next quarter-century nobody will be able to get a bachelor’s degree in mathematics without a passing familiarity with categories any more than one could avoid groups now, and it’s not just due to politicking on the part of its proponents as I’ve heard asserted.
First of all, categories are tremendously useful as a metamathematical language. I’ll show in the future how it unifies the First Isomorphism theorems, for example. I’ll also show how, in the language of categories, direct products of groups are like greatest lower bounds.
“So what,” the naysayer cries, “if this language says that those two concepts are related?” So, mathematics is about analogies. I can begin to understand this because I definitely understand that and this and that are similar in a certain way. Maybe knowing something about greatest lower bounds will tell me something new to look for in direct products of groups. Even if not, the relationships can help illuminate to newcomers — be they students or just lay readers — the essential points of the structures we consider, and more importantly why we consider them.
But there’s also another side of categories that the opposition completely ignores: a category can be just as useful a concrete mathematical structure as a group can, and the framework of categories can harmoniously sew together other objects into a coherent whole. The various rings and modules of matrices over a given field meld into the category of all matrices over that field. The braid groups weave together into the category of tangles.
And what do we gain from this categorical viewpoint? If unifying language isn’t enough for you, try this: category theory is, at its core, the language of the analytic/synthetic approach to mathematics in particular and all sciences in general. The scientific epistemology is to break complicated systems down into simpler parts, to understand those simple parts, and to understand how to reassemble them into the whole. This is exactly what category theory brings to the table: a systematic study of the nature of composition and how compositions transform when moving from one domain of discourse to another.
Category theory is the language of analogies, and analogies are the lifeblood of mathematics. Algebra gives us analogies between equations. Categories give us analogies between theories. Our future is concerned with analogies between analogies.
I’ve complained before about the opacity of the application procedure. I’ve gotten to the point now that I’m asking all outstanding schools what my status is. Following is a list — which I will keep updating — of departments that have closed their application procedures without bothering to inform applicants that they have been rejected. Petty? Maybe. But I’m the one who’s gotten screwed here, so the least they can do is be publicly known as not acting in good faith towards their applicants.
- George Mason University
- Kansas State University
- Northwestern University
- Purdue University
- Stony Brook University
- University of Arizona
- University of California, Davis
- University of California, Irvine
- University of California, Los Angeles
- University of Michigan
- University of Regina
- University of Tennessee
- University of Toronto
Note: I am giving departments the benefit of the doubt if they say they did send word. Still I am shocked — shocked — at how many letters the post office has lost.
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.]
Someone I know at one of the schools I applied to let me know that my application packet doesn’t have a teaching or research statement. Did I do something wrong? I went back to MathJobs and checked the application status form. It didn’t request them. I checked the job ad. Also didn’t request them.
Is this standard operating procedure, to tacitly require application materials not requested in the job ad? I’d noticed others that seemed to have rather thin requests for documents. Am I not hearing from them because I didn’t send them documents they didn’t ask for? Was I supposed to use telepathy to determine what they really wanted me to send?
I evidently can’t say “” in the title to a post. Today I’ve seen a lot of people talking about “ Day”, though, so I suppose I’m expected to do so as well. So I will.
I hate day. Hate hate hate hate hate this day. Hate it. Hate every simpering stupid vacant mathematics-insulting moment of it. Hate the sensibility that thinks anyone will like it. Hate the implied insult to mathematics by its belief that anyone would be entertained by it.1
Let me put it this way: why don’t we have a “The Sky Is Blue Day”? It’s about as obvious. The constant shows up everywhere for very good reasons, and once you’re really comfortable with it you’re just not that impressed. There has to be some constant satisfying any one of its definitions, and those all have to be the same thing. Beyond that, the decimal expansion — 3.1415926535… — is purely an accident of our notation for real numbers. When I see the same taped-glasses pedants who whine that the year 2000 is only an accident of base ten turn around and join in this silly fetishization of it’s incredibly depressing.
And then there’s the argument that this sort of thing acts as a springboard for mathematical interest. Listen carefully here: the only people who care about Day are already interested in at least applications of mathematics. That has such-and-so a value isn’t going to win any converts, and recitations of its decimal expansion make nonmathematicians think we’re all some sort of social outcasts who have nothing better to do with our time than that — if that’s what mathematics is, why bother going into such a dry and boring subject? Besides which, Lore Sjöberg said it best: “A value of pi that’s accurate to the 31st digit is good enough to measure the circumference of the entire universe within one proton, so anything beyond that is bordering on the mathsturbatory.”
Anyway, works the way it does for all sorts of good reasons, and it’s very well understood why. There are other constants with just as complicated expansions that we have very little idea why they work the way they do. Why not come up with a day for Feigenbaum’s constant?
So, mathematicians throw off the shackles of Day.
1 With apologies to Roger Ebert