There’s a quite remarkable article by Barry Mazur in the latest issue of the Bulletin of the AMS. It brings together ideas about elliptic curves and deformations of Galois representations that were used by Wiles to prove Fermat’s last theorem, mirror symmetry, quantization, non-commutative geometry and much more. I’m not convinced it all hangs together, but it’s a wonderful piece of expository writing.
Mazur claims to be inspired by a very interesting seminar held every week in the Harvard math department called the Basic Notions Seminar, parts of which have recently been put online. This issue of the Bulletin is dedicated to the great French mathematician Rene Thom, who died nearly two years ago. The articles by Michael Atiyah and Dennis Sullivan about Thom’s work in topology are well worth reading.
I happened to see this article and the following comes to mind:
a^p + b^p = c^p. If the p-norm, p of a metric space is not 2, then we have cyclotomic fields in a non-Hilbert space. Elliptic curves are just projections of SUSY spinors and are examples of cyclotomic fields. When p is 2, we can’t have cyclotomic fields and can have only simple harmonic functions. The interesting thing is that a Riemann manifold requires just one spinor to fix it’s position (like a rotating basketball), but when p>2, four spinors are needed and the space is quantized, by an integer ratio of a/c. Note that if ‘a’ is prime the field can’t be cyclotomic.
Wick ordering is a property of cyclotomic fields as well as NCG. My guess it that manifolds in non-Hilbert space are comprised of SUSY spinors and are exclusively solutions to non-linear PDEs. All non-Hilbert spaces are quantized — that’s a shock and Banach space is not a linear or complete vector space — interesting.
About the introduction of unstability in classical mechanics (page 22/328) I have from time to time redirecting people to Newton’s Principia, proposition 1, Book I, to read how Newton himself decides explicitly to carry any quantum of angular momentum to zero. A lot of people mistakes this theorem as a trivial one of angular momentum preservation, didn’t noticing that the continuum character of classical mechanics is defined there.
By the way, I remember to have read a preprint where someone uses the unstability concept not only to get “h” out of classical mechanics, but also to get “c” out of galilean group. It speaks of stable and unstable theories or so.