I don't know much about the equilibrium theory he won a Nobel prize for. But when I was 12 (in 1959) someone gave me Martin Gardner's "Scientific American Book of Mathematical Puzzles and Diversions," which had a chapter on The Game of Hex, which was invented independently by Piet Hein in 1942 and by John Nash in 1946. I have greatly enjoyed Hex ever since. Later, I learned the beautiful definition of an (abstract) smooth n-dimensional manifold in a college course. It was nice to learn that every manifold was in fact a submanifold of some Euclidean space R^p, in fact for some p <= 2n. Soon after that, we were taught the definition of an (abstract) Riemannian metric on a manifold, which was extremely satisfying and felt like exactly the right definition for making a manifold into a metric space. It was natural to wonder if every Riemannian manifold was also a submanifold of some Euclidean space, carrying the inherited Riemannian metric. Then someone told me that Nash had proved this. Originally he showed that for a smooth (C^oo) isometric embedding of a compact n-manifold into R^p, this was always possible for any p >= n(3n+11)/2. It's now known that at least for a compact smooth Riemannian n-dimensional manifold, it has an isometric embedding into R^q for any q >= n(n+1)/2 + max(2n, n+5). This compares with the earlier result of p >= n(3n+11)/2 = n(n+1)/2 + n(n+5). Maybe a more surprising result is that Nash originally proved that for a smooth manifold M^n (of differentiability class C^oo again) then for any dimensional Euclidean space R^s, s >= n+1, into which M has a smooth "short" embedding (one in which no distance is magnified), then: M has a C^1 *isometric* embedding in R^s. (The original theorem used R^(s+1), but N. Kuiper immediately showed this extra dimension was not necessary.) The strangeness of this result is exemplified by the familiar square torus R^2 / Z^2. This can be mapped via a short embedding to a small torus of revolution in R^3, and by the Nash-Kuiper theorem can be isometrically embedded C^1 into R^3. You probably won't be able to think of a way to do this, but recently an explicit such embedding has been found: < http://www.emis.de/journals/em/images/pdf/em_24.pdf >. ——Dan
On May 24, 2015, at 8:06 AM, Erich Friedman <erichfriedman68@gmail.com> wrote:
http://www.huffingtonpost.com/2015/05/24/john-nash-dead-a-beautiful-mind_n_7...