Lie group E8 mapped
 

I just saw this on Yahoo! news.

[quote]WASHINGTON (AFP) - After four years of intensive collaboration, 18 top mathematicians and computer scientists from the United States and Europe have successfully mapped E8, one of the largest and most complicated structures in mathematics, scientists said late Sunday.

Jeffrey Adams, project leader and mathematics professor at the University of Maryland said E8 was discovered over a century ago, in 1887, and until now, no one thought the structure could ever be understood.

"This groundbreaking achievement is significant both as an advance in basic knowledge, as well as a major advance in the use of large scale computing to solve complicated mathematical problems," Adams said.

He added that the mapping of E8 may well have unforeseen implications in mathematics and physics which won't be evident for years to come.

E8 belongs to so-called Lie groups that were invented by a 19th century Norwegian mathematician, Sophus Lie, to study symmetry.

The theory holds that underlying any symmetrical object, such as a sphere, is a Lie group.

Balls, cylinders or cones are familiar examples of symmetric three-dimensional objects.

However, mathematicians study symmetries in higher dimensions. In fact, E8 itself is 248-dimensional.

Today string theorists search for a theory of the universe by looking at E8 X E8.

The scientists said the magnitude of the E8 calculation invited comparison with the Human Genome Project.

While the human genome, which contains all the genetic information of a cell, is less than a gigabyte in size, the result of the E8 calculation, which contains all the information about E8, is 60 gigabytes in size, they said.

This is enough to store 45 days of continuous music in MP3-format. If written out on paper, the answer would cover an area the size of Manhattan.[/quote]

Source: [url]http://news.yahoo.com/s/afp/20070319/ts_alt_afp/ussciencemathematicsfrancegermany_070319121747[/url]

I thought that it was pretty interesting.

If you like that sort of thing:smile:

This is a better link with more details on the achievement:

[url=http://golem.ph.utexas.edu/category/2007/03/news_about_e8.html]A team of mathematicians led by David Vogan have computed the Kazhdan–Lusztig–Vogan polynomials for E[sub]8[/sub].[/url]

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...)

I didn't think that mapping E8 would be that hard.....

[quote=Uncwilly;101627]I didn't think that mapping E8 would be that hard.....[/quote]

Was Romania included?

Was watching a piece on this on [I]Baywatch[/I] 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 inter-toe 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.

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

[SIZE="1"](I'm being silly because I have no idea what E8 is all about, and the articles "explaining" it merely made my head ache)[/SIZE]

[quote=ewmayer;101632]Was watching a piece on [I]Baywatch[/I] 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.

[/quote]

Better rocket ships?? :w00t: :shock: :lol:

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.

[quote][SIZE=1](I'm being silly because I have no idea what E8 is all about, and the articles "explaining" it merely made my head ache)[/SIZE][/quote]
We thought it had something to do with chess.

:yawn:

[QUOTE=akruppa;101642] ... Alex

[SIZE="1"](I'm being silly because ... , and the articles "explaining" it merely made my head ache)[/SIZE][/QUOTE]

The popular articles seem to border on incoherence; and Baez's descriptions
(as in maxal's link) aren't typically known for keeping the pre-req'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 90-degree 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, n-by-n; two orthogonal groups,
one 2n-by-2n, 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 2n-by-2n 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, proto-typical.

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 1-complex 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 16-dimensions from E6 and the other
in 27-dimensions 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 upper-half space, in (complex) dimension n(n+1)/2, which
occurs as the space of parameters describing n-dimensional Abelian
varieties. Uhm, that's with 1-dim 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 p-th fourier coef counts the solns of the
eqn mod p (as in wiles's proof of Fermat, a counter-example 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
77-cpu hours (after four years real-time from 18 mathematicians). One
doesn't need to know much about Ventor's shotgun sequencing method,
or the rooms-full of (custom) alphas grinding away for months to confirm
that it's not-so-much the E8-computation 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: [url]http://www.nytimes.com/2007/03/20/science/20math.html[/url]

Alright, who's the wise guy? I think we can narrow it down quite quickly:

1) David Hasselhoff admirer - that's pretty much me, akruppa and Xyzzy;
2) Nothing better to do - that rules out akruppa;
3) Super-secret "invisible edit" forum privileges - uh huh, that's what I thought.

-Ernst, a.k.a. E5

[i]"Lie groups, damned Lie groups, and E8"[/i]

Ear shot!
 

[QUOTE=Xyzzy;101647]We thought it had something to do with chess.

:yawn:[/QUOTE]

:wink: I'm sorry! I'm hard of hearing - jet engines, Pratt and Whitney/Rolls Royce , you know !

yeah did you say chess or chest ? :grin:

Mally :coffee:

[QUOTE=ewmayer;101632]Was watching a piece on this on ...
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.[/QUOTE]

Glad to have the Baywatch episode info clarified (previous, more recent post).

I've gotten through to some of the more informative links; and have a much
better idea of what they think they were doing --- not sure you'll be happier.

Several of the links (from Baez's blog entry, maxal's post) give nice colored
pictures of the Dynkin diagram I was verbalizing about. First, on the "super"
part of the supercomputer computation, they had some help from Elkies -
otherwise they'd still be waiting for a "larger supercomputer". Bob might
be pleased to hear this one, the key idea seems to have been to run the
program four times to find the solution mod m, four moduli, then patched
the solution together via the Chinese remainder thm. I was just typing
"mod p, for primes", but the actual moduli used were 251 (prime), then
253 = (11)(23), then 255 = (3)(5)(17) and, finally 256 --- sure enough,
these are relatively prime, and Chin.Rem applies. So that 77 cpu hours
was distributed over 4 computations, just over 19 hrs each. The super
part appears to have been that they had 64Gb of ram to use --- ah,
on a "FusionA8" from Western Electric. Ooops, that was supposed to have
been "nearly 3 days solid processing" on the machine, with AMD chips.
Just checked with our grad student that has an account on the machine -
claim is that at the time of the computation Red Hat was only recognizing
8 of the 16 cpus (fixed since then, with a re-installation of the Red Hat).
That would be 3*24 = 72, so a bit more than 3 "solid days" times 8 cpus,
for 616 cpu hours, err ... four computations, average 154 cpu hours each.
Ah, yeah, 8 dual cores, "DDR2 667/533/400 Memory"; 8000 series.
With 10/100/1000 ethernet onboard "networking", infiniband interconnect
support. Sounding like super memory access/data exchange to people
more familiar? The 616 >= 3*24*8 amd cpu hours would be a slow day
for ecm here, otherwise.

Anyway, Adams did an update yesterday on "what we really did". Claim
is that they expect to get the data needed to be able to check an Arthur
conjecture on residues of Eisenstein series --- they still need to identify
which "perhaps 200 of the 453,060" satisfy Arthur's condition. -Bruce

ref: [url]http://www.liegroups.org/AIM_E8/technicaldetails.html[/url]