# E8 Structure Decoded

March 19, 2007 4:50 PM Subscribe

Math Team Solves the Unsolvable E8

"If you thought writing calculations to describe 3-D objects in math class was hard, consider doing the same for one with 248 dimensions. Mathematicians call such an object E

"If you thought writing calculations to describe 3-D objects in math class was hard, consider doing the same for one with 248 dimensions. Mathematicians call such an object E

_{8}, a symmetrical structure whose mathematical calculation has long been considered an unsolvable problem. Yet an international team of math whizzes cracked E_{8}'s symmetrical code in a large-scale computing project, which produced about 60 gigabytes of data. If they were to show their handiwork on paper, the written equation would cover an area the size of Manhattan."And it would fit handily on a $40 hard drive, with room to spare.

It's still a great achievement, I think. I'm not smart enough to know.

posted by chairface at 5:03 PM on March 19, 2007 [2 favorites]

It's still a great achievement, I think. I'm not smart enough to know.

posted by chairface at 5:03 PM on March 19, 2007 [2 favorites]

They knew the answer was in the back of the book, right?

posted by joe lisboa at 5:04 PM on March 19, 2007 [5 favorites]

posted by joe lisboa at 5:04 PM on March 19, 2007 [5 favorites]

*...calculations to describe 3-D objects...*

I love science reporting.

posted by DU at 5:10 PM on March 19, 2007

bardic, you totally beat me. I'll have to suffice with just saying....

MATH CLUB GEEKS!!!

posted by matty at 5:18 PM on March 19, 2007

MATH CLUB GEEKS!!!

posted by matty at 5:18 PM on March 19, 2007

*Math Team Solves the Unsolvable E8*

No-one has ever claimed it was "unsolvable", because describing an object as "unsolvable" is idiotic.

*...a symmetrical structure whose mathematical calculation has long been considered an unsolvable problem.*

No it hasn't. Of course, "calculating E8" is nonsense anyway. They did do something interesting, but you'll scarcely find a whiff of what it was in all this horrible writing.

*the written equation would cover an area the size of Manhattan."*

Their work isn't an "equation".

*E8 (pronounced "e eight")*

Thanks for clearing that up.

posted by Wolfdog at 5:20 PM on March 19, 2007

wait, so they don't have to show their work? That's the excuse I used when I copied someone else's homework back in fourth grade.

posted by jourman2 at 5:21 PM on March 19, 2007

*er...yes I did the homework, I just didn't show my work...*posted by jourman2 at 5:21 PM on March 19, 2007

This is interesting from the perspective of this quote, which I have often heard repeated:

"Beauty is the first test: there is no permanent place in this world for ugly mathematics." — G. H. Hardy (A Mathematician's Apology)

Of course, this isn't the first computer-assisted proof. The only proofs we have of the four color theorem are computer-aided for example.

posted by Rictic at 5:25 PM on March 19, 2007

"Beauty is the first test: there is no permanent place in this world for ugly mathematics." — G. H. Hardy (A Mathematician's Apology)

Of course, this isn't the first computer-assisted proof. The only proofs we have of the four color theorem are computer-aided for example.

posted by Rictic at 5:25 PM on March 19, 2007

The phrase "math whizzes" pretty much indicates that this article is devoid of content.

Also, the word "unsolvable" doesn't really apply here. They solved it, didn't they? Hence, not unsolvable.

Why does science reporting have to suck so much?

posted by number9dream at 5:27 PM on March 19, 2007

Also, the word "unsolvable" doesn't really apply here. They solved it, didn't they? Hence, not unsolvable.

Why does science reporting have to suck so much?

posted by number9dream at 5:27 PM on March 19, 2007

and for the love of God can we get a nerdporn tag here.

posted by jourman2 at 5:27 PM on March 19, 2007

posted by jourman2 at 5:27 PM on March 19, 2007

@number9dream - to be fair the author the "unsolvable" quote is pretty much a direct quote from one of the project heads.

posted by jourman2 at 5:31 PM on March 19, 2007

*"E8 was discovered over a century ago, in 1887, and until now, no one thought the structure could ever be understood," said Jeffrey Adams, project leader and a mathematics professor at the University of Maryland.*posted by jourman2 at 5:31 PM on March 19, 2007

Well, G. H. Hardy notwithstanding, E

Please recognize self-promotional blather when you see it. First of all, they "They Said It Couldn't Be Done!!!!" business is hooey, and second of all, "enumerating" is still a long way from "understanding."

They must have spent 1% of their budget on the 77 hours of supercomputer time and the remaining 99% on press releases. I'm getting email about this from every yahoo from here to kalamazoo. This is a big, remarkable feat of computation, but the language in the reporting is just outlandish.

posted by Wolfdog at 5:41 PM on March 19, 2007 [2 favorites]

_{8}exists, and 453,060 is how many representations it has. If you're going to make a list of them - and that's what they did - then that's how long it will be. And it will continue to exist, permanently, independent of anyone's aesthetic preferences.*"E8 was discovered over a century ago, in 1887, and until now, no one thought the structure could ever be understood," said Jeffrey Adams,*Please recognize self-promotional blather when you see it. First of all, they "They Said It Couldn't Be Done!!!!" business is hooey, and second of all, "enumerating" is still a long way from "understanding."

They must have spent 1% of their budget on the 77 hours of supercomputer time and the remaining 99% on press releases. I'm getting email about this from every yahoo from here to kalamazoo. This is a big, remarkable feat of computation, but the language in the reporting is just outlandish.

posted by Wolfdog at 5:41 PM on March 19, 2007 [2 favorites]

**Wolfdog**,

Since the links don't properly explain what's going on, would you mind laying it out for the idiot masses (like myself?) I'm not being snarky, I'd really like to understand what's being done here. You seem to be saying that all they did was make a list of representations of a mathematical construct?

posted by Sangermaine at 5:48 PM on March 19, 2007

To clarify, I get that E8 is an object with a huge number of dimensions, and that what the team did (if I understand correctly) is map out all of the different possible symmetries. I guess what I'm asking is, is there some special significance to this? It seems like they took a problem that's been around since 1887, but was too complex to work out completely before. Now that we have more powerful computers, a team worked it out. Is there some importance to E8 in general?

posted by Sangermaine at 5:58 PM on March 19, 2007

posted by Sangermaine at 5:58 PM on March 19, 2007

Decent explanation over at Good Math, Bad Math.

posted by elvolio at 6:14 PM on March 19, 2007 [3 favorites]

posted by elvolio at 6:14 PM on March 19, 2007 [3 favorites]

Speak for yourself!The only proofs we have of the four color theorem are computer-aided for example.

posted by Flunkie at 6:16 PM on March 19, 2007 [1 favorite]

*"If they were to show their handiwork on paper, the written equation would cover an area the size of Manhattan."*

If you don't show your work, you only get half-credit.

posted by mr_crash_davis at 6:27 PM on March 19, 2007

*Beauty is the first test...*

Agreed, in spades, which is why I am hoping with everything I have that if E8 is truly in some sense fundamental to our universe, it doesn't look quite so...Deadheadish?

posted by adamgreenfield at 6:27 PM on March 19, 2007

*Since the links don't properly explain what's going on, would you mind laying it out for [nonexperts] (like myself?)*

Please trust I'm being sincere when I say I believe that giving an accurate, accessible, and elementary explanation of this in the space of a few paragraphs on MetaFilter would be a vastly greater human achievement than what Adams, et al have done. I may come back and give a try at explaining what I can, but it will take some time & care.

posted by Wolfdog at 6:38 PM on March 19, 2007

I came up with the answer to this once when I was huffing nitrous but forgot it once I came to.

posted by The Straightener at 6:47 PM on March 19, 2007

posted by The Straightener at 6:47 PM on March 19, 2007

Math, is there anything it can't do?!

aside from getting me a girlfriend...

*sigh*

posted by The Power Nap at 6:55 PM on March 19, 2007

aside from getting me a girlfriend...

*sigh*

posted by The Power Nap at 6:55 PM on March 19, 2007

John Baez has the best explanation I've seen.

posted by gleuschk at 6:56 PM on March 19, 2007 [2 favorites]

posted by gleuschk at 6:56 PM on March 19, 2007 [2 favorites]

Lie groups are, in general, pretty terrible. I took a whole semester of a graduate level math class on manifolds in general, and not one of the grad students in the class came out understanding anything. All I know now is that mathematicians who study Lie groups are 1) actually physicists in disguise and 2) crazy.

posted by TypographicalError at 6:57 PM on March 19, 2007

posted by TypographicalError at 6:57 PM on March 19, 2007

*"Yet an international team of math whizzes cracked..."*

huh huh huh huh huh

he said "crack.."

huh huh huh huh huh

he said "whiz.."

huh huh huh huh huh

posted by ZachsMind at 7:01 PM on March 19, 2007

Ergh. The only thing I can make of this is that it has some connection to group theory, which I learned just enough of to get through P-chem. More power to those who get it (and those who solved it)!

posted by Joe Invisible at 7:13 PM on March 19, 2007

posted by Joe Invisible at 7:13 PM on March 19, 2007

And I love that the they used "RTFM" in their slides.

posted by Joe Invisible at 7:14 PM on March 19, 2007

posted by Joe Invisible at 7:14 PM on March 19, 2007

*Math Team Solves the Unsolvable E8*

No-one has ever claimed it was "unsolvable", because describing an object as "unsolvable" is idiotic.

No-one has ever claimed it was "unsolvable", because describing an object as "unsolvable" is idiotic.

Solvable Lie Group

but i'm pretty sure that's not what they meant...

a representation of a group is essentially a way of writing the group as a bunch of matrices (i.e. linear transformations on a vector space.)

posted by geos at 7:17 PM on March 19, 2007

*John Baez has the best explanation I've seen.*

posted by gleuschk at 6:56 PM on March 19

posted by gleuschk at 6:56 PM on March 19

i second that. i think this goes to show that with enough pr you can get the media to write about *anything*

by they way, John Baez and Joan Baez are cousins...

posted by geos at 7:21 PM on March 19, 2007 [2 favorites]

Math is pretty much at the point now where there is no point in writing laymen's explanations because there are no non-math analogies that apply.

It makes me wonder how new mathematicians ever get up to speed. Is it because they are super-specialized now?

posted by smackfu at 7:23 PM on March 19, 2007 [2 favorites]

It makes me wonder how new mathematicians ever get up to speed. Is it because they are super-specialized now?

posted by smackfu at 7:23 PM on March 19, 2007 [2 favorites]

In Jeff Adams' defense: he's a guy who thinks math is really cool, and he just completed a project that people in the world of math thought was really difficult, maybe impossible. I think he was just trying to convey the enormity and excitement of what he had done.

The Good Math, Bad Math link and the AIM press release both say that this result will have implications in quantum physics including string theory and supergravity. So it does have some usefulness in that aspect (as to whether you think string theory is useful or not, well I can't help you there).

Full disclosure: I took a complex analysis course with Jeff Adams last semester. He is a very fine lecturer, a great teacher, and a really nice guy.

posted by bluefly at 7:25 PM on March 19, 2007

The Good Math, Bad Math link and the AIM press release both say that this result will have implications in quantum physics including string theory and supergravity. So it does have some usefulness in that aspect (as to whether you think string theory is useful or not, well I can't help you there).

Full disclosure: I took a complex analysis course with Jeff Adams last semester. He is a very fine lecturer, a great teacher, and a really nice guy.

posted by bluefly at 7:25 PM on March 19, 2007

So, one reason this is interesting is that E8 is a simple group. Oh wait, I better explain what a group is. A group is a set with an operation defined on it, like the integers under addition. There are some further restrictions on how the operation interacts with the set in order for the thing to be a group, but I don't want to get into detail here.

There are an infinite number of groups out there. There are even an infinite number of groups of finite size. Some of these finite groups can be built up from smaller finite groups. (Again, I don't want to get into details.)

Now, if you think of this "building" operation as being similar to multiplication, simple groups bear the same relation to the set of all groups that the primes bear to integers. A simple group is atomic, in the sense that it is not built up from smaller groups, just as a prime number has no factors other than 1 and itself.

The crazy thing is that unlike the primes, mathematicians have been able to classify all the finite simple groups*. What I mean by classify is that they've set down what form a simple group may take. Most finite simple groups fall under a few categories. But there are a few others which defy such categories. And my understanding is that E8 is one of them. In fact, there are only a small number of these category-defying simple groups. And by small, I don't simply mean finite, but maybe like 100 or so, although 100 probably isn't the exact number.

So describing all the ways in which E8 can be represented has no direct analog to the prime numbers, but imagine if mathematicians discovered a formula which described most of the primes, in fact all but a few of them. Imagine further that the biggest prime not describable by this formula has, like, a million digits. If somebody actually bothered to figure out this number and compute it, it would be similar to what these guys did with E8, only it sounds like its biggest representation took up more space than just a million digits.

*Or so they think. Whenever they've found a bug in the proof of the classification, they've been able to fix it. But given that the proof takes up many thousands of pages, even most mathematicians have to take its validity on faith.

posted by A dead Quaker at 7:41 PM on March 19, 2007 [3 favorites]

There are an infinite number of groups out there. There are even an infinite number of groups of finite size. Some of these finite groups can be built up from smaller finite groups. (Again, I don't want to get into details.)

Now, if you think of this "building" operation as being similar to multiplication, simple groups bear the same relation to the set of all groups that the primes bear to integers. A simple group is atomic, in the sense that it is not built up from smaller groups, just as a prime number has no factors other than 1 and itself.

The crazy thing is that unlike the primes, mathematicians have been able to classify all the finite simple groups*. What I mean by classify is that they've set down what form a simple group may take. Most finite simple groups fall under a few categories. But there are a few others which defy such categories. And my understanding is that E8 is one of them. In fact, there are only a small number of these category-defying simple groups. And by small, I don't simply mean finite, but maybe like 100 or so, although 100 probably isn't the exact number.

So describing all the ways in which E8 can be represented has no direct analog to the prime numbers, but imagine if mathematicians discovered a formula which described most of the primes, in fact all but a few of them. Imagine further that the biggest prime not describable by this formula has, like, a million digits. If somebody actually bothered to figure out this number and compute it, it would be similar to what these guys did with E8, only it sounds like its biggest representation took up more space than just a million digits.

*Or so they think. Whenever they've found a bug in the proof of the classification, they've been able to fix it. But given that the proof takes up many thousands of pages, even most mathematicians have to take its validity on faith.

posted by A dead Quaker at 7:41 PM on March 19, 2007 [3 favorites]

Well, Max, I guess I'd see to it that none of you ever have to open another math book again for the rest of your lives.

posted by XQUZYPHYR at 7:45 PM on March 19, 2007 [1 favorite]

posted by XQUZYPHYR at 7:45 PM on March 19, 2007 [1 favorite]

and if it was written on paper cut from the worlds oldest tree, it would be on the oldest paper that ever been written on and possibly cover an area the size of bellshill.

posted by sgt.serenity at 8:10 PM on March 19, 2007

posted by sgt.serenity at 8:10 PM on March 19, 2007

I think the proof for solving E8 is more of a Shelbyville idea...

posted by mosk at 8:24 PM on March 19, 2007

posted by mosk at 8:24 PM on March 19, 2007

*the written equation would cover an area the size of Manhattan.*

If the Math Team were subjected to an atomic wedgie worthy of their stature, the stretched elastic band would reach halfway to the Moon.

posted by CynicalKnight at 8:37 PM on March 19, 2007 [2 favorites]

*Is there some importance to E8 in general?*

Mathturbation. Next up: E16!

*I may come back and give a try at explaining what I can, but it will take some time & care.*

Lemme guess... the MetaFilter margin isn't large enough to write your proof, right?

posted by Civil_Disobedient at 8:54 PM on March 19, 2007

Well, they computed a large matrix of polynomials by a standard procedure, with the small tweak of using modular arithmetic plus the chinese remainder theorem to make it all fit in RAM. It may be somewhat useful, but nothing revolutionary.

posted by metaplectic at 9:46 PM on March 19, 2007 [1 favorite]

posted by metaplectic at 9:46 PM on March 19, 2007 [1 favorite]

A dead Quaker's useful explanation is kind of like a layman's analogy, but more like "an analogy to make the doings of greater nerds understandable by lesser nerds". I wonder what we should call it.

posted by tehloki at 11:16 PM on March 19, 2007

posted by tehloki at 11:16 PM on March 19, 2007

*Most finite simple groups fall under a few categories. But there are a few others which defy such categories. And my understanding is that E8 is one of them.*

Well, for starters it's not a finite group.

*So describing all the ways in which E8 can be represented has no direct analog to the prime numbers, but imagine if mathematicians discovered a formula which described most of the primes, in fact all but a few of them. Imagine further that the biggest prime not describable by this formula has, like, a million digits. If somebody actually bothered to figure out this number and compute it, it would be similar to what these guys did with E8, only it sounds like its biggest representation took up more space than just a million digits.*

This is actually a pretty good analogy. The human genome analogy that's in the articles is also pretty good - and notice how the completion of the HGP hasn't suddenly rendered all the mysteries of human life and biology transparent. Having a big pile of data isn't the same as having insights.

*Solvable Lie Group... but i'm pretty sure that's not what they meant...*

Ha, I knew someone would pop up with the technical use of "solvable". Very clever. It has nothing to do with the everyday use of the word as in the articles, though, and even in the technical sense the negation of "solvable" is not "unsolvable".

Seriously, mathematicians just don't use the word "unsolvable" to describe problems that do in fact have a solution, and "find the character table of E8" is such a problem. There are things that are provably impossible (like trisecting the angle with ruler and compass). There are propositions that are formally undecideable based on a given set of axioms. But this isn't like that.

If you like to make analogies between Lie groups and finite groups, though, this brings the computational "understanding" of E8 up to about where the understanding of the Monster was, circa 1980. You can safely reckon there will still be plenty of surprising things to learn about it.

posted by Wolfdog at 11:51 PM on March 19, 2007

*Seriously, mathematicians just don't use the word "unsolvable" to describe problems that do in fact have a solution, and "find the character table of E8" is such a problem. There are things that are provably impossible (like trisecting the angle with ruler and compass). There are propositions that are formally undecideable based on a given set of axioms. But this isn't like that.*

How mathematicians use words is, how does the Lewis Carrol go: "as a mathematician, when i use a word it means exactly what i intend it to mean..."

How many card-carrying mathematicians are there in this thread, and what color is your card?

One of the vaguer things I find baffling as a non-algebra-ist is the difference between finite and infinite, say, lie groups; i tend to feel like they might as well be different theories once you get past that they are both examples of groups.

One thing a non-mathematician might wonder is: why would Plato make something as big and ugly as E8? What would it even mean to 'understand' it...

posted by geos at 5:20 AM on March 20, 2007

Few months ago our computer science professor was ready to bet that the E8 Structure can't be decoded. Well, now I have some news for him ;)

posted by techfreak at 5:42 AM on March 20, 2007

posted by techfreak at 5:42 AM on March 20, 2007

*What would it even mean to 'understand' it...*

Well, you know, size isn't everything. You think about the set of ALL 248x248 invertible matrices; that's also an infinite group; it contains E8 as a mere subset. Yet, the bigger set is conceptually simple; you understand it thoroughly, computationally, algebraically and to a certain extent even geometrically ("Hey, a matrix is just a thing that says the first basis vector goes HERE, and the second basis vector goes HERE, dot dot dot, and the 248th basis vector goes THERE." "Let's see how large the image of a unit cube is!" etc.) in any linear algebra course. At just a little more advanced level, you can understand the topological nature of the set - it's not compact, it's a nice covering of a Grassman manifold, and so on. And you can work out, in ways more clever or more elementary, what all the irreducible representations of it are. At some point, you feel you understand it, and that's it.

To make an analogy, everyone who's been through basic school algebra feels like they "understand" the x-y plane. It's infinite, but it's also not complicated. On the other hand, it contains within it sets like the Mandelbrot set, or various Cantor-type sets, or fantastic, wild curves, each with its own parcel of deep and complicated structural questions.

*why would Plato make something as big and ugly as E8?*

No reason to think it's intrinsically ugly; perhaps you just haven't yet seen the shadows from the proper angle for them to resolve into something lovely. On the other hand, I

*love*the existence of sporadic object in mathematics. I personally don't believe in a creator - but in the same way a weaver may deliberately introduce flaws in a rug in deference to the existence of some more perfect power, I think if there

*is*a creator, then sporadic objects could be poetically interpreted as the same sort of thing: deference to non-negotiable foundations that even the omnipotent must respect.

posted by Wolfdog at 6:09 AM on March 20, 2007 [3 favorites]

So, apparently you need a spirograph to solve the E8 problem?

posted by aught at 6:43 AM on March 20, 2007

posted by aught at 6:43 AM on March 20, 2007

*I personally don't believe in a creator - but in the same way a weaver may deliberately introduce flaws in a rug in deference to the existence of some more perfect power, I think if there is a creator, then sporadic objects could be poetically interpreted as the same sort of thing: deference to non-negotiable foundations that even the omnipotent must respect.*

This is an attractive point of view, but I personally tend more toward Israel Gelfand's thesis that the the sporadic finite simple groups (and, more generally, sporadic structures -- the "exceptions" to our grand classification theorems) are not really groups at all, but "objects from a still unknown

infinite family, some number of which [also] happened to be groups, just by chance."

In other words, you can look at Hawaii and say Hey look, it's 137 islands -- or you can reframe your view and say Hey look, it's a mountain range, and we were just seeing a few of the peaks. We don't have a concept yet for the "mountain range" that we're seeing the peaks of, but that's exciting, not limiting.

A grand example of the sort of thing I'm talking about is the whole circle of ideas surrounding the ADE classification. I came to these things from the representation theory of local rings, where the ADE singularities arise as the answer to a finiteness question. If you only know about that aspect of ADE, they seem sort of arbitrary; why these five critters (two infinite families and three exceptional guys) and not others? It's not until you see them come up in singularity theory, Platonic solids, and half a dozen other places that you realize their protrusion into representation theory is just an accident, an avatar of their overall amazingness, rather than an amazing thing in itself.

Oh yeah: one of those exceptional guys? It's E

_{8}.

posted by gleuschk at 6:53 AM on March 20, 2007 [1 favorite]

*It's not until you see them come up in singularity theory, Platonic solids, and half a dozen other places that you realize their protrusion into representation theory is just an accident, an avatar of their overall amazingness, rather than an amazing thing in itself.*

This is beautiful writing, and I understand what you're laying out in schema - the Hawai'i analogy helped, of course.

But please understand that I'm being sincere when I say you've lost me otherwise, through no fault of your own. It troubles me that, as a not-unintelligent, reasonably numerate adult, I couldn't follow

*any*of the "introductory" materials linked from this page.

I have no problem believing that, as gnarly as it is, E8 is somehow a fundamentally deep structure or quantity - see, I don't even know which descriptor is correct - but I do have trouble with the idea that something so fundamental would not ultimately be resolvable to something you don't need to be a postdoc to wrap your head around.

My prejudice has always been the interaction of simple parts is quite enough to create the most robust complexity, so why not here? Or is that just so embarrassingly and self-evidently the wrong question that I should just give up?

posted by adamgreenfield at 7:03 AM on March 20, 2007

*I personally tend more toward Israel Gelfand's thesis that the the sporadic finite simple groups (and, more generally, sporadic structures -- the "exceptions" to our grand classification theorems) are not really groups at all, but "objects from a still unknown infinite family, some number of which [also] happened to be groups, just by chance."*

Oh, I'm very sympathetic to that point of view, too, which is actually one reason I've been a bit vitriolic about the language in the press. I work with some pretty poorly-understood objects where the state of the art, I'm afraid, is still "Cool! We found another one!" But they will coalesce into a better and better picture with time.

Some things

*are*sporadic, though. S

_{6}is the only symmetric group with an outer automorphism, a "sporadic" phenomenon. Among spheres, only S

^{1}, S

^{3}, and S

^{7}admit homotopy-associative H-space structures. Alone among dimensions, 4-dimensional Euclidean space admits nonstandard differentiable structures. Aside from the even polygons, there is only one centrally-symmetric, self-dual regular polytope (in, mind you, my favorite dimension). I think these are all legit examples of sporadic phenomena that can't be "blamed" on a poorly-formed classification scheme, nor is there any need to. Of course, "sporadic" isn't a technical term here, so that's all debatable as a matter of taste.

posted by Wolfdog at 7:24 AM on March 20, 2007

*My prejudice has always been the interaction of simple parts is quite enough to create the most robust complexity, so why not here?*

That's exactly what's going on here. In this case, the interaction of a few very simple axioms are giving rise to structures that are not only exceedingly complex, but somehow

*irreducibly*complex. This means that they can't be obtained from other, smaller ones by just stapling them together. That's the rough meaning of the word "simple" in this context.

To expand on that a little, a

*group*is just a place where you can add and subtract things. The set of all fractions is a group -- add or subtract any two of them, you get another one, and the usual rules you learned in school (associativity, commutativity, etc) still hold. The positive integers, on the other hand, aren't a group, since you don't always get a positive integer back when you subtract two positive integers.

Once you strip away all the things that are special to fractions (like common denominators and all that), and just keep the rules, you've got the axioms for a group. Actually, it's more fruitful to toss out the commutativity rule too -- commutative groups turn out to be pretty boring. Now you can look around for other structures that satisfy those axioms. Turns out there are lots, some of which happen to be finite. The best example here is the numbers on a clock face. You can add any two of them, and you get another one back (if the result was bigger than 12, you just subtract 12 to get back down below 12).

I won't go further into what "simple" means technically, but you're not far off to think of it as "irreducible". The stunning thing is that there are "irreducible" finite groups of arbitrarily big size -- most of them fall into a few identifiable families, but there are 26 "sporadic" ones that don't fit into any families we understand.

Here's yet another link to an "introductory" description (yet again by Baez -- he's got this territory locked down as far as exposition goes), this time of the ADE stuff I was mentioning above. While you skim it, I'll think about whether there's any way to describe E

_{8}without using the word "Lie".

Wolfdog, I just learned that fact about the outer automorphisms of S

_{6}last month. You're right, that seems to be an honestly sporadic [and super-mega-weird, to my way of thinking] occurrence.

posted by gleuschk at 7:29 AM on March 20, 2007 [2 favorites]

I just realized why the phrase "irreducibly complex" rolled off my fingers so easily. Blech.

posted by gleuschk at 7:42 AM on March 20, 2007

posted by gleuschk at 7:42 AM on March 20, 2007

*I personally don't believe in a creator - but in the same way a weaver may deliberately introduce flaws in a rug in deference to the existence of some more perfect power, I think if there is a creator, then sporadic objects could be poetically interpreted as the same sort of thing: deference to non-negotiable foundations that even the omnipotent must respect.*

posted by Wolfdog at 6:09 AM on March 20

posted by Wolfdog at 6:09 AM on March 20

you are hedging your bets here. are you for or against "intelligent design" in mathematics? it's surprising how many mathematicians agape in horror at cavemen riding dinosaurs retreat to "platonic" mathematics in their own playground.

*I'll think about whether there's any way to describe E8 without using the word "Lie".*

but that's the thing, there is a heck of alot more structure, transcendental as it were, to a lie group. it seems like the 'lie' part far outweighs the group part. are they geometric objects that just happen to have a group structure, or algebraic objects that just happen to be manifolds?

posted by geos at 8:00 AM on March 20, 2007

*you are hedging your bets here.*

No, I'm not. It's perfectly logical, and healthy, to speculate about the potential consequences of a hypothesis even if you think it's not true.

*are they geometric objects that just happen to have a group structure, or algebraic objects that just happen to be manifolds?*

It's a floor wax

*and*a dessert topping. That's why we love them.

*there is a heck of alot more structure, transcendental as it were, to a lie group. it seems like the 'lie' part far outweighs the group part.*

No, not really. At least no more than the 'finite' part outweighs the 'group' part when you study finite groups. Everything in finite group theory is heavily colored by number theory and integer arithmetic, because, hey, there are

*n*elements in your group. That obviously doesn't get to play any role at all in Lie Group theory, but why should it?

posted by Wolfdog at 8:24 AM on March 20, 2007

If they've decoded the structure, I just want to know where the booths for Halo 3 and Penny Arcade are going to be.

...oh, E

posted by Uther Bentrazor at 8:42 AM on March 20, 2007

...oh, E

**8**... nevermind.posted by Uther Bentrazor at 8:42 AM on March 20, 2007

*No, I'm not. It's perfectly logical, and healthy, to speculate about the potential consequences of a hypothesis even if you think it's not true.*

you suggested that 'god' gave us the definition of the group: (your omnipotent resistant foundation? or perhaps i misunderstood)

*sporadic objects could be poetically interpreted as the same sort of thing: deference to non-negotiable foundations that even the omnipotent must respect*

i think maybe you mean that 'god' must respect conclusions based upon pure logic, but the definitions that make 'group theory' are entirely 'man-made' IMHO; they are convenient not fundamental.

*Everything in finite group theory is heavily colored by number theory and integer arithmetic, because, hey, there are n elements in your group. That obviously doesn't get to play any role at all in Lie Group theory, but why should it?*

when gleuschk muttered about Lie Groups without the Lie part, he was in part lamenting the fact that he has to deal with objects in the infinitely differentiable manifold category, the inherent structure being not readily interpretable algebraically so maybe if you are lucky you can shift categories and the algebra is clearer... or maybe you just shift categories and have to prove alot of weird little lemmas and never figure out whether you made any progress.

if you want and dessert and find yourself eating floor-wax it sucks. if you want to do algebra and find yourself worrying about whether something 'converges' it sucks.

posted by geos at 9:03 AM on March 20, 2007

*MATH WHIZZES COMPUTE E8, PROVE GOD!*

Overhead, one by one and without any fuss, the stars were going out.

posted by CynicalKnight at 9:16 AM on March 20, 2007 [3 favorites]

...if you want to do algebra and find yourself worrying about whether something ~~converges~~ is divisible by

Chacun à son goût. Me, I'm omnivorous.

posted by Wolfdog at 9:25 AM on March 20, 2007

*p*it sucks...Chacun à son goût. Me, I'm omnivorous.

posted by Wolfdog at 9:25 AM on March 20, 2007

So if I were to roll in here and talk about Hartry Field and nominalism, you guys would drag me out into the parkinglot and beat me with bagel toasters?

posted by The Power Nap at 12:31 PM on March 20, 2007

posted by The Power Nap at 12:31 PM on March 20, 2007

*So if I were to roll in here and talk about Hartry Field and nominalism, you guys would drag me out into the parkinglot and beat me with bagel toasters?*

posted by The Power Nap at 12:31 PM on March 20

posted by The Power Nap at 12:31 PM on March 20

or just ignore you. mathematicians don't care, if Pat Roberston took over the NSF they'd be just as happy to praise the lord for each new result. it's just one big video arcade and as long as someone is pumping in the quarters they are ready to play...

how's that for nominalism.

posted by geos at 3:57 PM on March 20, 2007

I still remember my 400-level Number Theory course, two decades ago, and how the prof told us that one of the applications of some elegant thingamajigger we'd spent a lot of time on was 'closest packing of spheres in 255-dimensional space' (if I remember it right).

I've used it for years now as an example of the Things I Don't Remember About My Major, which is odd, because there are probably thousands of other things which I quite literally don't remember.

posted by stavrosthewonderchicken at 5:24 PM on March 20, 2007

I've used it for years now as an example of the Things I Don't Remember About My Major, which is odd, because there are probably thousands of other things which I quite literally don't remember.

posted by stavrosthewonderchicken at 5:24 PM on March 20, 2007

Pardon the following. I don't have a blog (fuckwit) anymore, so I'm putting this here.

I've been reading lots of lamentation about how terrible all the popular-press articles about this result are, and it makes me a little sad. They

At the ICM last fall in Madrid, I attended a panel discussion titled

It turned out that du Sautoy was, as far as I can tell, almost single-handedly responsible for getting the Perelman story in the newspapers. He called the Guardian and, using his contacts there, bullied/cajoled/whatever them into running the first article about Poincaré and the possibility of Perelman turning down the Fields. By playing up the human aspect of the story (reclusive Russian genius, blah blah), he actually got them to run a moderately substantive story, with hints at the statement of the theorem and a smidge of history. Later articles tried to get deeper into the mathematics with varying results, from the atrocious ("He proved that donuts have no holes!") to the not-so-bad. But none of it would have happened if one guy hadn't made a few phone calls and insisted that this was news worth reporting.

Mathematicians do an almost universally lousy job of talking to the outside world. There are lots of explanations for this: discipline-wide autistic tendencies, not actually giving a damn about the outside world, intellectual elitism at its most rarefied, and the simple fact that adamgreenfield and Wolfdog pointed out above, that's it's just hard to explain exceedingly specialized fields, which have their own language, to someone who doesn't speak the language.

I'm conflicted about that last reason. Mathematics

On the one hand, this stuff is just

On the other hand, if we don't make the attempt, then we deserve to be mocked and scorned and de-funded. I think the NYT article, and all the other risibly awful articles about E

posted by gleuschk at 7:28 AM on March 21, 2007 [3 favorites]

I've been reading lots of lamentation about how terrible all the popular-press articles about this result are, and it makes me a little sad. They

**are**terrible, no doubt. But check it: they exist! Did you ever think you'd see an article in the New York Freakin Times featuring a photo captioned "Jeffrey D. Adams and a Lie group"? That's crazy shit, man.At the ICM last fall in Madrid, I attended a panel discussion titled

*Should mathematicians care about communicating to broad audiences?*. This was just around the time of the big Perelman dustup, which was everywhere in the news even before he declined the Fields Medal (two days before the panel). One of the panelists was Marcus du Sautoy, and what he had to say made a huge impression on me.It turned out that du Sautoy was, as far as I can tell, almost single-handedly responsible for getting the Perelman story in the newspapers. He called the Guardian and, using his contacts there, bullied/cajoled/whatever them into running the first article about Poincaré and the possibility of Perelman turning down the Fields. By playing up the human aspect of the story (reclusive Russian genius, blah blah), he actually got them to run a moderately substantive story, with hints at the statement of the theorem and a smidge of history. Later articles tried to get deeper into the mathematics with varying results, from the atrocious ("He proved that donuts have no holes!") to the not-so-bad. But none of it would have happened if one guy hadn't made a few phone calls and insisted that this was news worth reporting.

Mathematicians do an almost universally lousy job of talking to the outside world. There are lots of explanations for this: discipline-wide autistic tendencies, not actually giving a damn about the outside world, intellectual elitism at its most rarefied, and the simple fact that adamgreenfield and Wolfdog pointed out above, that's it's just hard to explain exceedingly specialized fields, which have their own language, to someone who doesn't speak the language.

I'm conflicted about that last reason. Mathematics

*is*much less accessible, from a purely linguistic point of view, than things like medicine, physics, and chemistry are. We learn words like "molecule", "radiation", and "DNA" in grade school, but nothing about manifolds or groups. (Instead we perpetuate the confusion between "mathematics" and "arithmetic", but that's another rant.) Why is this? Is it a problem?On the one hand, this stuff is just

**hard**. There are plenty of pure-mathematics PhDs -- algebraists, even! -- who don't have the dimmest clue what a Kazhdan-Lusztig-Vogan polynomial is. There are plenty of PhD programs that don't even offer a*course*in Lie theory. (Disclosure: I got my own degree from one of those departments, and I am one of those algebraists.) We're talking here about material that takes years*beyond*a PhD to understand, and the vast majority of people who do similar things for a living don't put that time in. To take an intelligent, numerate, motivated person from zero-knowledge to the frontier of what's going on just isn't a reasonable request.On the other hand, if we don't make the attempt, then we deserve to be mocked and scorned and de-funded. I think the NYT article, and all the other risibly awful articles about E

_{8}, are fantastic, if only because*someone*is trying to get the word out. The word is that we're working on hard problems over here, that we're pushing the boundaries of what we though pure cogitation was capable of, that we're making progress. We're making connections between things that don't seem connected, finding underlying reasons why different-looking things are similar. Some of those things are going to change our understanding of the physical world, be it subatomic or prehistoric; some are going to give us better JPEG compression algorithms; some are going to decrease turbulence around a stent in someone's aorta; and some are just achingly beautiful. And lots -- most, probably -- isn't applicable to "real-world" problems at all ... at least not so far as we can see today. But tomorrow we'll see farther.posted by gleuschk at 7:28 AM on March 21, 2007 [3 favorites]

*achingly beautiful*...

I get that.

Now imagine that you can appreciate that beauty as something approaching an object in its ownself, and I can only grasp its shadow on a 2D plane. It hurts to be conscious of that gap, just as it's occasionally caused me pain to know that there are colors I can't see.

Anyway, thanks for sharing your eloquent thoughts with us. It helps. Some. : . )

posted by adamgreenfield at 1:55 PM on March 21, 2007

Oh, and I just got to the bottom of the PDF deck from the first comment. I'll warn you that the very last slide hits like a hammer. Or maybe I'm just vulnerable. Anyway, I shed a tear.

posted by adamgreenfield at 2:14 PM on March 21, 2007

posted by adamgreenfield at 2:14 PM on March 21, 2007

Yeah, so, you can take your pick on this one: either 'achingly' was me getting carried away by my own rhetoric, or it's a technical mathematical term that I couldn't possibly explain in such a small textarea.

To try to tone the persiflage down a skosh, I have to say that as far as I can tell, this result about E

I spent the drive home thinking about theorems or theories that I would really call 'achingly beautiful'. I couldn't think of many, but the classification of finite subgroups of SO

Also, I had gotten lost on page ~102 of that PDF deck and quit before the end -- you're right, the last page does hit like a brick.

posted by gleuschk at 3:37 PM on March 21, 2007

To try to tone the persiflage down a skosh, I have to say that as far as I can tell, this result about E

_{8}doesn't qualify as beautiful, achingly or otherwise. It's a very difficult computation about an object that we don't really understand, which object (in its various avatars) does come up in beautiful contexts. It's one more step toward understanding that object; it doesn't (as the NYT article would have you believe) lay bare "the deep inner structure of the universe" or anything of the sort. Incremental progress is the name of the game here.I spent the drive home thinking about theorems or theories that I would really call 'achingly beautiful'. I couldn't think of many, but the classification of finite subgroups of SO

_{3}sure is amazingly pretty (another technical term), and (shock!) E_{8}shows up here again, as the symmetry group of the icosahedron. Anyway, thanks for a pleasant drive.Also, I had gotten lost on page ~102 of that PDF deck and quit before the end -- you're right, the last page does hit like a brick.

posted by gleuschk at 3:37 PM on March 21, 2007

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