Re: This Week's Finds in Mathematical Physics (Week 223)
- From: john_ramsden@xxxxxxxxxxxxxx
- Date: Sun, 18 Dec 2005 18:39:38 +0000 (UTC)
tessel@xxxxxx wrote:
>
> On Sun, 27 Nov 2005 john_ramsden@xxxxxxxxxxxxxx wrote:
>
Many thanks for your detailed and interesting reply.
> When geometric algebra and linear algebra battle to win the hearts and
> minds of applied mathematicians, there is no real contest, because overall
> the benefits of sticking close to notions used in by -far- the greater
> portion of the literature vastly outweigh considerations of elegance. It
> may not be fair, but -it is so-, and in the end that's what matters most.
So I suppose we're stuck with the Gibbs "div" and "curl" formalism
for the foreseeable future?
Seems a shame, because it's not even as if geometric algebra is
at all hard to master, quite the reverse.
Clearly this inertia is the main impediment, as you say, although
I did wonder if Doran et als' "Euclidean spacetime" theory, which
they express in the geometric algebra formalism, might not have
slightly tainted the latter in the minds of physicists who perhaps
have zero tolerance of the slightest deviation from conventional
tenets of GR, especially a return to Euclidean notions (although
I hasten to add I'm not myself qualified to judge, and they do
say their theory agrees with GR in every prediction so far made
and observed).
> If you are a math/physics/CS student interested in getting the most bang
> for your buck, I'd have to urge learning about computational algebraic
> geometry (starting with basic Groebner basis computations with ideals in
> polynomial rings) in favor of learning about geometric algebra. I suspect
> that the latter is likely to remain a comparatively small if intruiging
> field for the foreseeable future, whereas computational algebraic geometry
> is rapidly becoming a standard component of the toolkit of every
> mathematically literate computer user. Adapt or die--- it's Nature's way!
I'd recommend "An Introduction to Grobner Bases" by Ralf Froberg,
1997, ISBN 0 471 97442 0.
Also, in support of your point about learning modern algebraic
geometry, I think that when general relativity is developed,
as Einstein and Schrodinger did in the late 1940s, using only
parallel connections rather than the usual metric approach,
the potential function turns out to be a polynomial. This
suggests that modern algebraic geometry as it relates to
algebraic varieties ("polynomials") may be more applicable
to physics than many people realize. (I must admit to being
somewhat hazy about the details though, and it may be this
doesn't make as much sense as I hope it does ;-)
.
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