On Nash’s embedding theorem. A little eulogy.

There are two kinds of mathematical contributions — work that is important to the history of mathematics and work that is simply a triumph of the human spirit. Paul Halmos.


Reference: A Beautiful Mind by Sylvia Nasar.


“The discussion of manifolds was everywhere,” said Joseph Kohn in 1995, gesturing to the air around him. “The precise that Ambrose asked Nash in the common room one day was the following: Is it possible to embed any Riemannian in a Euclidean space?”

It’s a “deep philosophical question” concerning the foundations of geometry that virtually every mathematician — from Riemann and Hilbert to Elie-Joseph Cartan and Hermann Weyl — working in the field of differential geometry for the past (19th century) had asked himself. The question first posed explicitly by Ludwig Schlaffi in the 1870’s, had evolved naturally from a progression of questions that had been posed and partly answered beginning in the mid-nineteenth century. First mathematicians studied ordinary curves, then surfaces, and finally, thanks to Riemann, a sickly German genius and one of the great figures of nineteenth century mathematics, geometric objects in higher dimensions. Riemann discovered examples of manifolds inside Euclidean spaces. But in the early 1950’s interest shifted to manifolds partly because of the large role that distorted space and time relationships had in Einstein’s theory of relativity.

Nash’s own description of the embedding problem in his 1995 Nobel autobiography hints at the reason he wished to make sure that solving would be worth the effort. “This problem, although classical, was not much talked about as an outstanding problem. It was not like, for example, the four colour problem.

Embedding involves portraying a geometric object as — or bit more precisely making it a subset of — some space in some dimension. Take the surface of a balloon. You can’t put it on a blackboard, which is a two-dimensional space. But you can make it a subset of a space of three or more dimensions. Now, take a slightly more complicated object, say a Klein bottle. A Klein bottle looks like a tin whose lid and bottom have been removed and whose top has been stretched around and reconnected through the side to the bottom. If you think about it, it’s obvious that if you try it in a three-dimensional space, the thing intersects itself. That is bad from a mathematical point of view because the neighbourhood in the immediate vicinity of the intersection looks weird and irregular, and attempts to calculate various attributes like distance or rates of change in that part of the object tend to blow up. But put the same Klein bottle in a space of four dimensions and the thing no longer intersects itself. Like a ball embedded in three-space, a Klein bottle in four-space becomes a perfectly well-behaved manifold.

Nash’s theorem stated that any kind of surface that embodied a special notion of smoothness can actually be embedded in Euclidean space. He showed that you could fold the manifold like a silk handkerchief, without distorting it. Nobody would have expected Nash’s theorem to be true. In fact, everyone would have expected it to be false. “It showed incredible originality,” said Mikhail Gromov, the geometer whose book Partial Differential Relations builds on Nash’s work. He went on:

“Many of us have the power to develop existing ideas. We follow paths prepared by others. But most of us could never produce anything comparable to what Nash produced. It’s like lightning striking. Psychologically, the barrier he broke is absolutely fantastic. He has completely changed the perspective on partial differential equations. There has been some tendency in recent decades to move from harmony to chaos. Nash says chaos is just round the corner. “

John Conway, the Princeton mathematician who discovered surreal numbers and invented the game of life, called Nash’s result “one of the most important pieces of mathematical analysis in this century.”

It was also, one must add, a deliberate jab at then fashionable approaches to Riemannian manifolds, just as Nash’s approach to the theory of games as a direct challenge to von Neumann’s. Ambrose, for example, was himself involved in a highly abstract and conceptual description of such manifolds at the time. As Jurgen Moser, a young German mathematician, who came to know Nash well in the mid-1950’s put it, “Nash didn’t like that style of mathematics at all. He was out to show that this, to his mind, exotic approach was completely unnecessary since any such manifold was simply a submanifold of a high dimensional Euclidean space.”

Nash’s more important achievement may have been the powerful technique he invented to obtain his result. In order to prove his theorem, Nash had to confront a seemingly insurmountable obstacle, solving a certain set of partial differential equations that were impossible to solve with existing methods.

That obstacle cropped up in many mathematical and physical problems. It was the difficulty that Levinson, according to Ambrose’s letter, pointed out to Nash, and it is a difficulty that crops up in many, many new problems — in particular, non-linear problems. Typically, in solving an equation, the that is given is some function, and one finds estimates of derivatives of a solution in terms of derivatives of the given function. Nash’s solution was remarkable in that the a priori estimates lost derivatives. Nobody knew how to deal with such equations. Nash invented a novel iterative method — a procedure for making a series of educated guesses — for finding roots of the equations, and combined it with a technique for smoothing to counteract the loss of derivatives.

Donald Newman described Nash as a “very poetic, different kind of thinker.” In this instance, Nash used differential calculus, not geometric pictures, or algebraic manipulations, methods that were classical outgrowths of nineteenth century calculus. The technique is now referred to as the Nash-Moser theorem, although there is no dispute that Nash was the originator. Jurgen Moser was to show how Nash’s technique could be modified and applied to celestial mechanics — the movement of planets — especially for establishing the stability of periodic orbits.

Nash solved the problem in two steps. He discovered that one could embed a Riemannian manifold in a three-dimensional space if one ignored smoothness. One had, so to speak, to crumple it up. It was a remarkable result, a strange and interesting result, but a mathematical curiosity, or so it seemed. Mathematicians were interested in embedding without wrinkles, embedding in which the smoothness of the manifold could be preserved.

In his autobiographical essay, Nash wrote:

So, as it happened, as soon as I heard in conversation at MIT about the question of embeddability being open I began to study it. The first break led to a curious result about the embeddability being realizable in surprisingly low-dimensional ambient spaces provided that one would accept that the embedding would have only limited smoothness. And later, with “heavy analysis,” the problem was solved in terms of embedding with a more proper degree of smoothness.

Nash presented his initial, “curious” result in a seminar in Princeton, most likely in the spring of 1953, at around the same time that Ambrose wrote his scathing letter to Halmos. Artin was in the audience. He made no secret of his doubts.

“Well, that is all well and good, but what about the embedding theorem?” said Artin. “You will never get it.”

“I will get it next week.” Nash shot back.

Jacob Schwartz, a brilliant young mathematician at Yale was also working on it concurrently.

Later, after Nash had produced his solution, Schwartz wrote a book on the subject of implicit function theorems. He recalled in 1996:

I got half the idea independently, but I couldn’t get the other half. It’s easy to see an approximate statement to the effect that not every surface can be easily embedded, but that you can come arbitrarily close. I got that idea and I was able to produce the proof of the easy half in a day. But then I realized that there was a technical problem. I worked on it for a month and couldn’t see any way to make headway. I ran into an absolute stone wall. I didn’t know what to do. Nash worked on the problem for two years with a sort of ferocious, fantastic tenacity until he broke through it. “

Week after week, Nash would turn up in Levinson’s office, much as he had in Spencer’s at Princeton. He would describe to Levinson what he had done and Levinson would show him why it didn’t work. Isadore Singer, a fellow Moore instructor recalled:

“He’d show the solutions to Levinson. The first few times he was dead wrong. But he didn’t give up. As he saw the problem get harder and harder, he applied himself more, and more, and more. He was motivated just to show everybody how good he was, sure, but on the other hand, he didn’t give up even when the problem turned out to be much harder than expected. He put more and more of himself into it.”

There is no way of knowing what enables one man to crack a big problem, while another man, also brilliant, fails. Some geniuses have been sprinters who have solved problems quickly. Nash was a long distance runner. If Nash defined von Neumann in his approach to the theory of game, he now took on the received wisdom of nearly half a century. He went into a classical domain where everybody believed that they understood what was possible and not possible. “It took enormous courage to attack these problems,” said Paul Cohen, a mathematician at Stanford University and a Fields medalist. His tolerance for solitude, great confidence in his own intuition, indifference to criticism — all detectable at a young age, but now prominent and impermeable features of his personality — served him well. He was a hard worker by habit. He worked mostly at night in his MIT office — from ten in the evening until 3.00 AM — and on weekends as well, with, as one observer said, “no references but his own mind” and his “supreme self-confidence.” Schwartz called it “the ability to continue punching the wall until the stone breaks.”

The most eloquent description of Nash’s single-minded attack on the problem comes from Jurgen Moser:

“The difficulty that Levinson had pointed out, to anyone in his right mind, would have stopped them cold and caused them to abandon the problem. But Nash was different. If he had a hunch, conventional criticism did not stop him. He had no background knowledge. It was totally uncanny. Nobody could understand how somebody like that could do it. He was the only person I ever saw with that kind of power, just brute mental power.”

On the nature of math — abstraction

Abstraction is such a central part of modern mathematics that one forgets that it was not until Frechet’s 1906 thesis that sets of points with no a priori underlying structure (not assumed points in or functions on \Re^{n}) are considered and given a structure a posteriori (Frechet first defined abstract metric spaces). And after its success in analysis, abstraction took over significant parts of algebra, geometry, topology and logic.


Nalin Pithwa.

Wisdom of Hermann Weyl w.r.t. Algebra

Important though the general concepts and propositions may be with which the modern and industrious passion for axiomatizating and generalizing has presented us, in algebra perhaps more than anywhere else, nevertheless I am convinced that the special problems in all their complexity consitute the stock and core of mathematics, and that to master their difficulties requires on the whole hard labour.

—- Prof. Hermann Weyl.