20070320, 09:53  #1  
Bemusing Prompter
"Danny"
Dec 2002
California
2,351 Posts 
Lie group E8 mapped
I just saw this on Yahoo! news.
Quote:
I thought that it was pretty interesting. 

20070320, 10:44  #2 
"Lucan"
Dec 2006
England
194A_{16} Posts 
If you like that sort of thing

20070320, 18:49  #3 
Feb 2005
FC_{16} Posts 
This is a better link with more details on the achievement:
A team of mathematicians led by David Vogan have computed the Kazhdan–Lusztig–Vogan polynomials for E_{8}. Last fiddled with by maxal on 20070320 at 18:49 
20070320, 19:24  #4 
Feb 2007
2^{4}×3^{3} Posts 
didn't have the time to follow your links, but I wonder whether this has an impact on the superstring GUTs (I was into that in an earlier life...)

20070321, 14:12  #5 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
9202_{10} Posts 
I didn't think that mapping E8 would be that hard.....

20070321, 14:44  #6 
"Lucan"
Dec 2006
England
2·3·13·83 Posts 

20070321, 16:11  #7 
∂^{2}ω=0
Sep 2002
República de California
2^{6}×181 Posts 
Was watching a piece on this on Baywatch last night, in which one mathematician familiar with the work was blathering about all the fantastic "practical applications" for this result. When he got to "better rocket ships" (and yes, he literally made that claim), I almost blew hot tea out my nose.
I was hoping at least for a "cure for cancer" (remember the SSC proponents pulling that one out?). Or failing that, relief for that annoying foot fungus responsible for most cases of itchy intertoe skin cracking. Neat result, but c'mon folks, no need to pull phony "practical applications" out of your collective rumps just to please the news media. It's embarrassing to watch. 
20070321, 17:28  #8 
"Nancy"
Aug 2002
Alexandria
2,467 Posts 
Well, maybe E8 will at least let them make some progress on flying cars at long last. They've promised us flying cars, what, fourty years ago? Now look out of your window: do you see any flying cars out there? I sure don't!
Alex (I'm being silly because I have no idea what E8 is all about, and the articles "explaining" it merely made my head ache) 
20070321, 18:08  #9  
Nov 2003
2^{2}·5·373 Posts 
Quote:
I can conceive of one practical application: What was actually computed was all of the different *representations* of the group. Such representations can be useful in error correcting codes. 

20070321, 18:09  #10  
"Mike"
Aug 2002
5·7·227 Posts 
Quote:


20070322, 16:23  #11  
Jun 2005
lehigh.edu
2^{10} Posts 
Quote:
(as in maxal's link) aren't typically known for keeping the prereq's to a minimum. If one knew what E8 is, then the description of the project to which the computation belongs as "calculating the representations of the split real form" might serve to locate the area to which the computation belongs. Or scrolling down a bit further, there's an extract of a reply to the blog entry from one the principal investigators (Jeffery Adams, at Maryland) which refers to a paper tiltled "Algorithms for representation theory of real reductive groups". Otherwise, ... Maybe it would be worth starting with the fact that the E in E8 refers to the fact that it occurs as an Exception. Actually, the largest of five exceptions: G2, F4, E6, E7 and E8. The numbers 2, 4, 6, 7 and 8 refer to the "rank", for which the easiest picture gives that number of dots in the corresponding Dynkin diagram. As an undergrad back at UofO I was assigned to read through a paper of Dynkin as a project in a summer NFS program between my freshman and sophmore year, since the paper only required linear algebra. There were several subsequent passes before I had a better idea of what the rank is. But anyway, the diagram for E8 consists of 7 linked dots along a line, then one more dot downward at a 90degree angle connected to the dot that's the 3rd from the last dot (on the right). The diagrams for E6 and E7 are similar, except for having 5, resp. 6, linked dots on the main line. These groups occur as exceptions in the theorem that says that every simple group is a classical group (one of four infinite families), with these five exceptions. So that's the linear group, nbyn; two orthogonal groups, one 2nby2n, the other (2n+1)by(2n+1), consisting of matrices with columns that are perpendicular vectors of length 1; and one more with columns satisfying a similar condition, but using the hermitian dot product  the 2nby2n sympletic group. The theorem's supposed to say that there are only these four classical matrix groups; but that's not correct, there are those five exceptions. A typical classification theorem, prototypical. I'm used to hearing about these exceptions as first occuring in E. Cartan's investigation from the 20's, as applied in functions of several complex variables. If we're used to functions of 1complex variable f(z) being periodic not only under z > z+1, which gives a fourier series, and fourier coef, but also under z > (1/z); then we could look for functions f(z1, z2, ..., zn) with a larger group of periods. Cartan's classification says that the only cases that occur are for functions on the complex domains associated with the four infinite families of classical groups, with just two exceptions, one in 16dimensions from E6 and the other in 27dimensions from E7. The other 3 exceptions, including E8, give real domains which can't be used to get complex domains. The most familiar examples are the domains associated with the sympletic group, the Siegel upperhalf space, in (complex) dimension n(n+1)/2, which occurs as the space of parameters describing ndimensional Abelian varieties. Uhm, that's with 1dim Abelian varieties being known as elliptic curves (the EC in ECM, yes?); for which the ones given by equations with rational coef are uniquely associated with the complex functions f(z) for which the pth fourier coef counts the solns of the eqn mod p (as in wiles's proof of Fermat, a counterexample would have given a curve that didn't correspond to any f(z)). To get back to the popularization of the new E8 computation, the NYtimes article reports that the supercomputer portion of the calculation took 77cpu hours (after four years realtime from 18 mathematicians). One doesn't need to know much about Ventor's shotgun sequencing method, or the roomsfull of (custom) alphas grinding away for months to confirm that it's notsomuch the E8computation that's large, as the amount of data needed to decribe the result of the computation: 1 Gbyte for the genome, which Baez describes as a "pickup truck full of books", as compared to 60 Gb to store the E8 answer, a "453,060 × 453,060 matrix of polyn". Or the NYtimes refers to as having computed 200 billion numbers. I stopped attempting to follow the links in Baez's blog entry just shortly before developing headache of my own. Bruce NYtimes ref: http://www.nytimes.com/2007/03/20/science/20math.html 

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