Here are some updates (mini blog) on the paper
A Sard theorem for graph theory
Here are some animations
August 27, 2015: on the local version [PDF], started to
make improvements (smaller typos and fixing inconsistency to label f2 the
ground state or confusion with writing the gradient additively or multiplicatively.
The discrete Nash embedding question has to be refined a bit. Of course, it is possible to
embed a finite simple graph G in a sufficiently large complete graph Kn
by taking n as the number of vertices of G and so, one can also embed it into refinements
of Kn. The question is whether there
exists a k which only depends on d, such that any d graph can be embedded in a
suitable Barycentric refinement Gm of Kk.
Since the degree of G can be arbitrary large, we can not put a bound on m.
The dependence of k as a function of d is known in the classical case, so that one can
expect something similar in the discrete.
I don't see any direct relation to the Nash embedding question for Riemannian manifolds
except for the analogy that both questions depend are of a metric nature and not only on the
topology and that in some sense, barycentric refinements are of a flat Euclidean nature.
Exciting would of course be to solve the discrete case (or a
variant of it) and verify that such a discrete result implies in a limit the continuum
result. The Nash embedding theorem is still not teachable in the sense that the proof is
too complicated to be presented in a class.
Arthur Sard (1909-1980)
grew up in New York, was an undergrad and grad student at Harvard University (graduated in 1931)
(masters in 1932) and PhD in 1936. The Sard theorem was his thesis
under guidance of Marston Morse. As the Genealogy
shows, he seems have been the last PhD student of Morse (who himself got his PhD at Harvard under the guidance of
George D. Birkhoff). His paper is
Sard lived later in Switzerland in
Binningen near Basel.
The Sard theorem is also called Morse-Sard theorem. Its easy to believe that this relates to Marston Morse, the PhD dad
of Sard, but it is Anthony Morse who proved a one-dimensional
version in 1939, got his pHD in 1937 at Brown and is the PhD father of Herbert Federer.
I found no picture of Sard on the web, but located a picture in the book "Multivariate Approximation Theory II,
(1982). Here is a scan:
Here is an obituary article in those proceedings: