# John Nash's Mathematical Legacy

It would seem that John Nash and his wife Alicia died tragically in a car accident on May 23, 2015 (reference). My condolences to his family and friends.

Maybe this is an appropriate time to ask a question about John Nash's work which has been on my mind for awhile. John Nash's best known work to the world at large involves his contributions to game theory, but to many geometers his work on embeddings of Riemannian manifolds is really his crown jewel. An excerpt from a note by Gromov:

When I started studying Nash’s 1956 and 1966 papers (it was at Rokhlin’s seminar ≈1968), his proof has stricken me as convincing as lifting oneself by the hair. Under a pressure by Rokhlin, I plodded on, and, eventually, got the gist of it... Trying to reconstruct the proof and being unable to do this, I found out that my ”formalization by definitions” was incomplete and my argument, as stated in 1972 was invalid (for non-compact manifolds). When I simplified everything up and wrote down the proof with a meticulous care, I realized that it was almost line for line the same as in the 1956 paper by Nash - his reasoning turned out to be a stable fixed point in the ”space of ideas”! (I was neither the first nor the last to generalize/simplify/improve Nash, but his proof remains unrivaled.)

So I'm wondering if anyone can comment on the legacy of Nash's work in geometry today. Have his ideas been absorbed into a larger theory? Have his techniques found applications outside of manifold embeddings?

Perhaps this is a good place to comment on other parts of his mathematical legacy as well, if anyone would like to.

• I flagged for CW so that we can upvote. I am sorry to hear the news – Benjamin Steinberg May 24 '15 at 16:03
• That's an impressive quote from Gromov. Elsewhere (ams.org/notices/201003/rtx100300391p.pdf) there is another impressive quote: "Raussen and Skau: This means that you read Nash’s work and were impressed by it very early? Gromov: Yes, I read it very carefully. And I still believe I am the only person who read his papers from the beginning to the end. By judging what people have written about it afterwards, I do not think they have read it." (Hat tip to user5831 here: mathoverflow.net/a/60137/2926) – Todd Trimble May 24 '15 at 16:36
• I'm wondering if anyone could comment on Nash's work after winning the Nobel Prize? It seems he was trying to do something with logic, but I wasn't able to get the gist of it. – Todd Trimble May 24 '15 at 17:13
• It is interesting to note that there was actually an error in Nash's original proof of the embedding result, as Nash discusses on his webpage: web.math.princeton.edu/jfnj/texts_and_graphics/Main.Content/… – Dan Ramras May 24 '15 at 20:48
• @ToddTrimble Yes, that's a nice interview. Later in the interview, Gromov's also says this nice quote (still talking about Nash): At least, his work in geometry was contrary to what everybody would expect, concerning the results, the techniques, the ideas he used. He did various matters in an extremely simple way, so that everybody could see it but nobody would believe it could work. – Kimball May 25 '15 at 8:05

Nash's major contributions, as far as I know, are the following:

1. His work on game theory. EDIT: This is viewed by many mathematicians as being more important to economics than mathematics. However, see the answer by Gil Kalai (someone else whose views should be taken much more seriously than mine).

2. His famous work on the existence of smooth isometric embeddings of Riemannian manifolds into Euclidean space. As Denis Serre mentions, he developed in this paper what is now known as the Nash-Moser implicit function theorem, which has been used in other applications.

3. His work on regularity of solutions to elliptic and parabolic PDE's, which were also obtained, I believe independently by Moser and DiGorgi. This is perhaps his most cited work.

4. His theorem about how any Riemannian manifold has a $C^1$ isometric embedding as a codimension 2 submanfold of Euclidean space. Kuiper showed that the embedding could be actually a hypersurface (codimension 1). This is perhaps his most spectacular theorem. Recently, De Lellis, László Székelyhidi, and their collaborators have used Nash's original "twist" construction to obtain new results on Onsager's conjecture, which is about the apparently totally unrelated topic of fluid dynamics.

Nash's influence on mathematics (and on my own work) is enormous.

• Sorry. It's already in more or less a random order. I also hope everyone has the sense to take anything said by Gromov far more seriously than by me. – Deane Yang May 24 '15 at 23:26
• In terms of keywords (since I haven't seen it mentioned on this page yet), the Nash-Kuiper theorem is one of the examples that led to the development of the h-principle. (Of course, those who would recognize this keyword probably would also know about Nash's contributions, so maybe it is not that key to include it...) – Willie Wong May 26 '15 at 7:53
• @WillieWong, as someone who has often heard the term 'h-principle', but didn't know that it was inspired by Nash (among others), I'm very glad you mentioned it! – HJRW May 26 '15 at 9:28

Nash's lesser known papers also contain some clever constructions. Here's an example:

A path space and the Stiefel-Whitney classes. Proc. Nat. Acad. Sci. U.S.A. 41, 320–321 (1955) [MR71081]

There, he provides a rather simple proof of the fact that Stiefel-Whitney classes of the tangent bundle $TM \to M$ of a smooth manifold $M$ do not depend on the smooth structure of $M$ (so that two manifolds $M \cong_{top} M'$ that are homeomorphic but not necessarily diffeomorphic have tangent bundles $TM$ and $TM'$ with identical Stiefel-Whitney classes). Apparently the result was earlier proven by Thom, though by completely different methods, with Nash's proof being much simpler. Nash introduced a homotopy theoretic model for $TM$ by representing $T_xM$ by the space of all continuous curves that start at $x$ but never come back to it. Since this model only uses the topology of $M$, the independence of $TM$, and its topological invariants, from the differential structure of $M$ follows.

See Appendix B of Hughes & Ranicki's Ends of Complexes (CUP, 1996) for more context and how this idea has been incorporated into further work on differentiable and topological manifolds.

Concerning Deane Yang's 4th point, permit me to cite the earlier MathOverflow question, "$C^1$ isometric embedding of flat torus into $\mathbb{R}^3$," which displays (and links to1) some amazing images from the Hévéa2 project's illustration of the Nash-Kuiper $C^1$ embedding theorem applied to the flat torus:

1 Courtesy of Benoît Kloeckner.
2 The Institut Camille Jordan, the Laboratoire Jean Kuntzmann, and the Grenoble Gipsa-Lab.

Why focus only on geometry? Nash investigated a lot of fields, in only thirteen (?) published papers. That on the isometric embedding contains a fixed point theorem that is now called the Nash-Moser iteration, which turns out to be extremely fruitful in many parts of analysis, especially in PDEs, when ordinary methods fail. More precisely, it is used when the linearization about a given solution yields small divisors troubles.

Nash also had a seminal work on compressible viscous fluid flows.

Edit. As mentionned by Pedro, one speaks of the Nash-Moser implicit function theorem. Nevertheless, the method is really a (far reaching) adaptation of Newton's iteration (as Pedro quote it). Isn't Newton method a fixed point algorithm designed to find and approach zeroes of functions ?

• I'm not familiar with Nash's work on fluid flows. Do you have a reference for this? – Deane Yang May 24 '15 at 17:03
• I found the paper: Le problème de Cauchy pour les équations différentielles d'un fluide général. (French) Bull. Soc. Math. France 90 1962 487–497. The abstract seems to indicate that there were some errors in the paper? Any chance you could say more about this paper? – Deane Yang May 24 '15 at 17:53
• There is an English translation of the fluid paper at www.math.cornell.edu/~bterrell/nash.ps that I made for my own study; it is not a scholarly translation but might be of use to someone. – Bob Terrell May 24 '15 at 22:17
• @EdwardDunne: About half of these 25 papers are reprints. – Dmitri Pavlov May 25 '15 at 6:58
• The Nash-Moser iteration scheme is not a fixed-point method, but rather an inverse / implicit function theorem. It is a sort of regularized Newton scheme. The regularization procedure along each iteration resembles (in fact, predates) renormalization group ideas. – Pedro Lauridsen Ribeiro May 25 '15 at 13:50

Let me mention Nash's two famous contributions to game theory:

The notion of equilibrium of non cooperative games and a proof that every game has an equilibrium.

Nash's bargaining model, his proposed axioms that a solution should satisfy, and his solution. Links: (Nash's paper; a good wiki page (Hebrew))

Addition: Nash also independently invented in 1947 the "game of HEX" (invented frst by Piet Hein in 1942, and he discovered the "stealing strategy" argument proving the player I has a winning strategy. (in a non-constructive way.)

More Let me copy my answer regarding Nash's work from a different question on the importance of Nash's work in game theory and economics.

The Nash equilibrium concept and the accompanying existence theorem is among the very few cornerstones in mathematical modeling of our reality .

Let me add a few more links. A paper by Ariel Rubinstein entitled "John Nash the master of economic modeling" and a document based on a seminar entitled: "The work of John Nash in Game theory." This includes the following quote from an earlier paper by Aumann.

Nash equilibrium is probably the main reason for the "game-theoretic revolution" in theoretical economics.

• It is probably worth remarking that this work has other broader cultural implications, at least according to the way Adam Curtis tells the story in his documentary The Trap. John Nash is interviewed in the first episode. At the same time, the frustrating thing is that quite a few people who would not have heard of John Nash otherwise have seen this documentary, and on the basis of Curtis's storytelling, construe Nash to be the progenitor of Thatcherism. – Joe Corneli May 29 '15 at 13:31

The Essential John Nash (ed. Kuhn and Nasar) contains the full text of nine of Nash's papers along with some editorial introductions and an autobiographical essay by Nash.

• Could someone give a clue as to what the Parallel Control chapter in that collection is about? – Igor Khavkine May 24 '15 at 21:28
• @IgorKhavkine: The paper is basically a list of desiderata for a parellel computer. – Steven Landsburg May 24 '15 at 23:04
• @StevenLandsburg, I've never heard of that aspect of his work before. Is there any significance to it? – Igor Khavkine May 24 '15 at 23:47
• @IgorKhavkine: I'm the wrong person to ask. – Steven Landsburg May 25 '15 at 15:03
• It's amazingly prescient, from a computer science perspective. Here's a link to the paper: rand.org/content/dam/rand/pubs/research_memoranda/2008/… – Martin M. W. May 26 '15 at 1:57

Although this is not directly about Riemannian geometry, his paper Arc structure of Singularities (1995) certainly classifies as sufficiently geometrical. By all accounts, this paper was actually conceived some 30 years or more earlier. Ex post facto, the contents of that paper were way ahead of their time.

In some twenty years, much has been written (and will be written) about the Nash problem for arc spaces of singular varieties.

Nash's paper "Real algebraic manifolds", Ann. of Math 56(1952), 405-421, started off quite a bit of work in real algebraic geometry, where Nash functions is a well-established term for $\mathcal{C}^\infty$-semialgebraic functions.

I would like to supplement others' very nice answers with this beautifully written article by Hamilton on the Nash-Moser theorem. It describes several nice applications of the theorem at the end, including:

• Existence of embeddings of surfaces with curvature bounds
• Existence theorems for the shallow water equations
• If $S$ is a submanifold of a Riemannian manifold $X$ then the space of submanifolds near $S$ with the same volume is a codimension $1$ submanifold of the space of submanifolds
• Nearby symplectic/contact structures can be conjugated via a diffeomorphism (up to cohomological restrictions)
• The diffeomorphism group of a compact manifold is a principal bundle over the space of smooth probability measures

I'm not sure if Nash himself actually proved any of these results, but I would include them as part of his legacy. It would seem that the Nash-Moser theorem (and its generalization, the h-principle, as someone else mentioned) is the larger theory into which Nash's ideas have been absorbed.

Interestingly, Nash's original embedding theorem for Riemannian manifolds was not presented as an application of the Nash-Moser theorem. Does anyone know if the former actually follows from the latter, or do they just have similar proofs?

• A few comments: a) First, there are two Nash isometric embedding theorems. One is the existence of a smooth isometric embedding into a Euclidean space of sufficiently high dimension. The other is the existence of a $C^1$ embedding or immersion into a Euclidean space of low codimension. – Deane Yang May 29 '15 at 0:42
• b) Nash never formulated the Nash-Moser implicit function theorem explicitly. He concocted an extremely ingenious iteration scheme to prove the first isometric embedding theorem. Moser showed how Nash's scheme implies an implicit function theorem for Frechet spaces, generalizing the standard one for Banach spaces. Hamilton created an even more abstract framework for the theorem, introducing the concept of smooth tame maps and estimates. – Deane Yang May 29 '15 at 0:43
• c) The h-principle is the abstract version of the arguments used in Nash's second isometric embedding theorem. As far as I know, it has nothing to do with the Nash-Moser implicit function theorem. However, see arxiv.org/abs/1111.2700 for a explanation why I might be wrong about this. – Deane Yang May 29 '15 at 0:47