Re: This Week's Finds in Mathematical Physics (Week 223)
- From: tessel@xxxxxx
- Date: Tue, 29 Nov 2005 08:44:45 +0000 (UTC)
On Sun, 27 Nov 2005 john_ramsden@xxxxxxxxxxxxxx wrote:
> For example I gather a famous mathematician called Gian Carlo Rota
Gian-Carlo Rota
> was working on what he felt was an essential revamp of
[geometric algebra]
> in the 1990s shortly before he unfortunately died.
>
> N.B. Geometric algebra shouldn't be confused with (modern) algebraic
> geometry, which is the theory of schemes developed by Grothendieck.
^
the modern foundation for
But there is a lot more to algebraic geometry than the notion of schemes!
Indeed, some classic subjects have undergone dramatic rebirths in recent
decades, and I suspect these trends were of more interest to Rota than
hyperabstractions arising from the desire for maximal generality. In
particular, "combinatorial algebraic geometry" (Groebner bases,
elimination theory, determinantal calculus, and such like) and
"computational invariant theory" seem to me to be close to his some of his
strongest research interests.
Anyone curious to learn what these subjects are about can consult various
books coauthored by Bernd Sturmfels. And see Rota's expository paper on
invariant theory for some trenchant comments on how unresolved issues from
the past can still bite us today (in this case, misunderstanding of
Cayley's "omega process" confused some nineteenth century leaders of
invariant theory, with consequences which can still be seen). Then, to
close the circle, see
http://www.math.uiuc.edu/Macaulay2/Book/
which is on computational algebraic geometry but includes a chapter on
teaching schemes using Macaulay2.
> Geometric Algebra for Physicists
> Chris Doran & Anthony Lasenby
> CUP
"Gtr": Doran has used geometric algebra to find an important coordinate
chart for the Kerr vacuum generalizing the Painleve chart for the
Schwarzschild spacetime:
Chris Doran, "A new form of the Kerr solution"
Phys. Rev. D 61 (2000): 067503
gr-qc/9910099
Every serious student of gravitation should be familiar with this chart.
BTW, nowadays symbolic computation engines like Maple often use concepts
allied to Groebner bases to solve systems of differential equations. For
example "casesplit" can "triangularize" systems of linear PDEs; this is
analogous to using Gaussian reduction in elementary linear algebra.
Keywords include: "differential rings", "Ore algebra", "Weyl algebra", and
"differential Galois theory". While I am on the subject, many CS topics,
such as the way that Maple represents knowledge internally, join
operatations in relational databases work, etc., afford nice examples of
important absract constructions from category theory, so abstraction need
not be impractical.
Computations involving ideals in differential rings are analogous to
Groebner basis computations involving ideals in polynomial rings, except
that we replace our polynomial rings with Ore algebras, which include
various algebras useful for the kind of "quantum calculus" computations
which have been discussed in previous Weeks. This is because these Ore
algebras can be interpreted as rings of operators, such as differential or
shift operators or q-deformed variants of these. This is in turn related
to things like automatic proof of identities involving special functions
such as Gauss's hypergeometric functions, which are of course very useful
for solving differential equations.
All users of applied mathematics should know something about this powerful
machinery, but it seems that there is still no undergraduate textbook
explaining it. Unfortunately, Ore algebra computations are trickier in
various ways than polynomial ring computations, but rapid progress is
making all this increasingly practical for daily use.
BTW, Ore's Ph.D. students included both Grace Hopper of computer fame and
Marshall Hall, whose Ph.D. students included Donald Knuth of algorithms
fame. Rota's Ph. D. students included Richard Stanley of enumerative
combinatorics fame. It would be difficult to overestimate the influence
of Rota and Ore on contemporary mathematics.
(Randomly selected trivia item: at the opposite extreme from the
beneficial pedagogical activity of someone like Rota, which former Ph.D.
student of Leo Kadanoff and Patrick Billingsley has achieved remarkable
success in his strenuous efforts to -dismantle- science education in the
U.S. over the past decade? Barf! Mumble mumble nonmetaphorical religious
war... mumble mumble belief in the void ... mumble mumble avian flu...
mumble mumble lethal stupidity...)
Er, we seem to be getting far afield. IIRC, JB -has- addressed your
question in past posts, but to offer my own answer:
There is an age-old creative tension in mathematics between the desire to
express some body of ideas in the "most natural" way (perhaps at the
expense of climbing to a higher level of abstraction), and the desire to
reduce things to the "most elementary" notions (which means attempting a
descent to sea level--- as any mountaineer will tell you, this is a far
more hazardous undertaking than climbing upwards!).
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.
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!
"T. Essel"
.
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