I was just reading online about the E8, 248-dimensional math problem being solved. I read several of the links provided, but got no closer to understanding what the hell the 248 dimensional problem is, how it was solved, and why it was important to solve. Is there a way to explain it in laymen’s terms so that a math-challenged person (such as I) can understand it?
Got any links, so we can take a gander at it?
Here’s the link:
This page and the ones it links to provide more detail: http://aimath.org/E8/
The comparison to the human genome project is good. What these guys have done, basically, is found the DNA for E[sub]8[/sub], which is the largest finite exceptional Lie group. Of course, that brings us to the question of what an exceptional Lie group is.
A course in modern algebra will begin with an introduction to the concept of a group, which is a structure that has some notion of multiplication of its elements. The multiplication has three properties:
- For any group elements a, b and c, a*(bc) = (ab)*c.
- There is an identity element; some e such that ae = ea = a for any group element a.
- For any group element a, there is a group element b with ab = ba = e.
In very rough terms*, a Lie group is a group where the multiplication is differentiable, just like all the functions you ran into back in calculus.
About 100 years ago, it was discovered that all but 5 finite Lie groups can be completely described by one of four categories and a positive integer. The other 5 are known as the exceptional Lie groups, and E[sub]8[/sub] is the biggest of them.
*A Lie group also has to be a differentiable manifold, but we’ll all be happier if I don’t get into what that means.
How does this compare to the classification of all finite simple groups? It sounds similar, in that each problem is about characterizing all possible groups of a certain sort. In both cases, there were several sporadic or exceptional cases, and the complete classification of these things involved identifying all these sporadic or exceptional cases.
Is this just solving a math problem for its own sake, or does this stuff have any practical applications? I looked this up, and I’d like to say that it looks Greek to me, but even Greek is more comprehensible to me. So I’m genuinely interested in what difference this makes to anybody other than mathematicians.
From what I’ve read, theoretical physicists seem to be getting pretty excited about it. But I’m no expert.
I don’t know about this problem in particular, but group theory has many practical applications, like cryptography and error-correcting codes. Modern digital communications and storage devices are heavily dependent on error-correcting codes.
You forgot one. A group also has to be closed, which is to say that, if a and b are in the group, then ab must also be in the group. For instance, the set [0.1,10] under standard real multiplication meets the other three requirements (associative, includes an identity, and includes inverses), but it’s not a group, since 1010 = 100, and 100 is not a member of the group.
And I haven’t personally encountered this group in my work, but I’d be willing to bet that the theoretical physicists who are getting excited about it are all string theorists. Anyone else would shudder at the thought of working with a mathematical object that big.
Closure is generally covered by defining the multiplication operator as an element of G X G -> G, and I didn’t think it was absolutely essential to spell it out for the high-level introduction I was going for.
Is there a pill that a guy can take to be brainy like you? It’s not fair. :smack:
Hmmm…so apparently there are no laymen’s terms, eh? Not sure I’m closer to understanding, but thanks for giving it a try.
This phrase need expansion for complete clarity to me.
The article quotes mathematician Marcus de Sautoy, thusly:
I hate to admit it, but this really can’t be broken down to completely elementary terms. As a rough lead-in, start with what ultrafilter said about groups and then read what I wrote today as a preparatory. I cover a rough idea of what makes a Lie group, what leads from there to Lie algebras, and what a representation is.
I’m talking with Jeff Adams and our advisor (Jeff was his first student, I’m his most recent) and trying to get a rough sketch I can understand of the K-L-V polynomials, which I’ll digest and report on on my above-linked weblog.
Actually the page that GFactor linked to above were quite helpful in gaining a basic understanding.
True, but there may be others reading the thread who are interested in a low-level but not superficial explanation. That’s what I’m giving over at my place.
So… basically say you took a line. That’s 1 dimension. A square is 2 dimensions. A cube is 3 dimensions. A cube through time is 4 dimensions. This is basically an object of 248 dimensions?
Because if not, I have no idea what the heck you guys are talking about. My eyes glazed over about 2 sentences into ultrafiler’s post