[Beowulf] The recently solved Lie Group problem E8

[Robert G. Brown]

On Thu, 22 Mar 2007, Peter St. John wrote:

> Well to me that's the point. My brain is too small for 500Kx500K matrices
> over a ring of 22 degree polynomials, too. So we throw a 16-node computer at
> it and crush it under the hobnailed jack-boots of Higher Mathematics.
> I wish I know more about the SAGE (machine) that hosts the SAGE (software)
> that was used for this, but apparently washington.edu's web server can't
> handle the CNN exposure as well as their number cruncher can crunch numbers.
> They are still down.

In the case of E8, using a computer is probably necessary, although one
would require a wetware interface to make the slightest bit of "sense"
out of the results anyway.

I do find this trend depressing, though.  Fermat's lost theorem proven
using computing -- nothing elegant, just crush it underfoot, as you say.
E8.  Next we'll hear that the Goldbach Conjecture is finally proven by
virtue of solving 10^17 specific cases and exploiting a proof that once
you have all of those cases proven you can iterate the result to
infinity somehow, or we'll hear of the Riemann Hypothesis being solved
this way -- nothing elegant, nothing that is (actually) of any USE.  We
all know that these are true anyway, at least as well as we know that
the theory of gravitation is true.  In neither case can they be proven
(yet, in the case of the math, never in the case of gravity), in both
cases they are known beyond any reasonable doubt via induction (see
Polya's lovely books on induction and mathematical reasoning).  Proving
these things by computer adds nothing to this -- in addition to the near
impossibility of actually judging the computational results (deep bugs
remaining a ubiquitous possibility in ALL complex computer code) which
always leaves a sliver of doubt even then, that doubt (expressed as
Jaynesian/Bayesian "degree of belief" on an information
theoretic/entropic basis) is already so small that it hardly changes on
a log scale from having done an exhaustive computation.

In the SPECIFIC case of E8 that isn't quite true.  Since string theory
as a theory of everything (TOE) may be covered by E8, and since string
theory is reportedly insanely complex and so big that exploring it to
find the RIGHT decomposition into whatever \times SU(whatever) by hand
might take lifetimes, it is barely possible that being able to enumerate
it even electronically will permit a systematic search to be performed
that can eliminate huge blocks of the possibilities and home in on what
we can at least HOPE is a small set of decompositions.  Ideally a
single, unique decomposition.

That would actually be pretty cool.  For the first time in pretty much
forever, we'd have an actual CANDIDATE TOE, and yet another important
step in "the end of physics" will have occurred.  (And note well the
quotes, please -- I'm not suggesting that physics research will come
close to ending with a TOE, only that it will finally have a firm known
basic foundation.

    rgb

> Peter
>
>
> On 3/22/07, Robert G. Brown <rgb at phy.duke.edu> wrote:
>> 
>> On Wed, 21 Mar 2007, Peter St. John wrote:
>> 
>> > Times have sure changed; with Wiles and Fermat's Last Theorm in
>> newspapers
>> > for over a year, then "A Beautiful Mind" from Hollywood; it's almost not
>> > surprising that the solution of a difficult math problem is mentioned at
>> > CNN.com.
>> >
>> > The Exceptional Lie Group E8 computation just got done (some info at
>> > http://www.aimath.org/E8/computerdetails.html about the details of the
>> > computation itself). Reference to the system SAGE is a bit ambiguous;
>> it's
>> > the name of a symbolic mathematics package and apparently also a 16-node
>> > system at the same University of Washington. Natually I was curious
>> about
>> > the computer, but ironically, it seems that while they can handle a
>> matrix
>> > with half a million rows and colums each (and each entry is a polynomial
>> of
>> > degree up to 22, with 7 digit coeficients), their departmental web
>> server
>> > can't handle the load of all of CNN's readership browsing at once :-)
>> >
>> > The group E8 itself, together with some explanation of the recent news,
>> is
>> > in wiki, http://en.wikipedia.org/wiki/E8_%28mathematics%29
>> >
>> > Dr Brown might explain better than I could how sometimes the best way to
>> > understand a thing is to break it down into simple groups of symmetries.
>> 
>> Don't you be puttin' that off on me now.  I get off of that particular
>> bus somewhere around the SU(N) stop, with rare excursions over into
>> point groups on the other side of the tracks.  Unitary, yes.
>> Orthogonal, why not.  SL(2,C) even.  Strictly UNexceptional.
>> 
>> > Apparently, one of the funky things about E8 is that the "easiest way to
>> > understand it" is itself.
>> 
>> Yeah, and like I have a brain that can manage ~500,000x500,000
>> complicated polynomial objects.  Thanks, I think... but not.
>>
>>    rgb
>> 
>> >
>> > Peter
>> >
>> 
>> --
>> Robert G. Brown                        http://www.phy.duke.edu/~rgb/
>> Duke University Dept. of Physics, Box 90305
>> Durham, N.C. 27708-0305
>> Phone: 1-919-660-2567  Fax: 919-660-2525     email:rgb at phy.duke.edu
>> 
>> 
>> 
>

-- 
Robert G. Brown	                       http://www.phy.duke.edu/~rgb/
Duke University Dept. of Physics, Box 90305
Durham, N.C. 27708-0305
Phone: 1-919-660-2567  Fax: 919-660-2525     email:rgb at phy.duke.edu