Quote:
Originally Posted by Tobin Dax
And Chris, thanks for the correct calculation. I was working too fast last night.
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But you got us started on some pretty interesting stuff.
I guess only KenG read my
thread on
Schubert calculus and how
enumerative geometry relates to
superstring theory (I haven't actually gotten to that last part yet!).
As an example of the interconnectedness of mathematics, you probably wouldn't guess that the stuff I discuss there about
Grassmannians has any relationship with
quantum calculus, but it does.
Search
John Baez's column
This Weeks Finds for a long riff on
reflection groups,
Coxeter/Dynkin diagrams,
semi-simple Lie groups,
parabolic subgroups,
cohomology of
flag manifolds, and more, in which among many other things he used quantum calculus to compute the vector space strucuture of the cohomology ring of a flag manifold (generalization of a Grassmannian). John has discussed many wonderful things, but this was my all time favorite :smile:
(I've been trying to get him to explain the generalization of what I said about the ring structure to general flag manifolds...)
To search his website: try the nifty search bar at upper right on his
home page. Years ago, I learned from Dornfest et al.
Google Hacks, O'Reilly, 3rd Edition, 2006 how to make a search bar, and passed the trick on to John. My one contribution to mathematical physics :wink:
Fourier transforms: this wonderful topic has been greatly generalized to
harmonic analysis, a topic which unifies many ideas from
representation theory,
operator theory, and
theory of PDEs. Applications extend to
dynamical systems and
analytic number theory. "Harmonics" appear as eigenfunctions of certain linear operators on function spaces (often, partial differential operators such as the Laplace operator), and the general idea is to decompose general functions in our function space into a "sum" of harmonics.
Just a tiny hint of how HUP arises naturally in pure math: it turns out that in operator theory, two natural operators are
multiplication by x, written
f -> M_x f and
differentiation wrt x,
f -> D_x f. These two operators do not commute:
D_x M_x f = f(x) + M_x D_x f for all functions, so we have
D_x M_x - M_x D_x = I
where
I is the identity operator. This is the starting point for learning about the Weyl algebra.
(In
symbolic dynamics this is reformulated using
shift operators, which suggests, correctly, a connection with
tiling dynamical systems such as the space of
Penrose tilings.)
Some suggested reading:
For the formulation of the HUP as Just Another Inequality in
L^p spaces :wink:
see problem 32, section 6.3, in my favorite real analysis text:
Gerald B. Folland,
Real Analysis,
Wiley, 1984.
See two old and fairly brief posts by myself on two vast, vast topics:
and see references therein.
A Wikipedia article I only skimmed but it seems OK (the anon is from UofC) is
Weyl algebra. Did I mention that the cohomology of Grassmannians has something to do with polynomial rings?
An old but superb exposition of harmonic analysis is
Kenneth I. Gross,
On the Evolution of Noncommutative Harmonic Analysis.
American Mathematical Monthly, Vol. 85, No. 7, 525-548. Aug. - Sep., 1978.
Prof. Gross won the
Chauvenet Prize for this paper. Those of you with access to a university library should be able to easily find it on your library shelves, or if your uni is wealthy, at
JSTOR. An earlier Chauvenet Prize paper, by Guido Weiss, is a fine introduction to complex methods in fourier transforms.
A fascinating expository post by--- no, darnitall, there be cranks here, I don't want them to go spam his blog with woo. I'll try to remember to PM the link.
Victor Kac and Pokman Cheung.
Quantum Calculus.
Springer, 2002.
Richard Kane,
Reflection Groups and Invariant Theory.
CMS, 2001.
See
this stillborn thread if you want to discuss the sampling theorem and what information theory says about HUP.
Cover and Thomas,
Elements of Information Theory.
Wiley, 1991.
(The best of dozens of fine textbooks on information theory, IMO, if only because it gives some impression of the vast reach of Shannon's creation.)