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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.

8I flagged for CW so that we can upvote. I am sorry to hear the news – Benjamin Steinberg – 2015-05-24T16:03:09.117

1In response to a comment now deleted: only mods can make questions CW, but the author can still do it for answers. – Joonas Ilmavirta – 2015-05-24T16:16:39.003

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That's an impressive quote from Gromov. Elsewhere (http://www.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: http://mathoverflow.net/a/60137/2926)

– Todd Trimble – 2015-05-24T16:36:47.18712I'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 – 2015-05-24T17:13:42.247

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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: http://web.math.princeton.edu/jfnj/texts_and_graphics/Main.Content/Erratum.txt

– Dan Ramras – 2015-05-24T20:48:27.3601I hope somebody with more expertise chimes up and summarizes Nash's work on cooperative games (I had the fortune to attend a talk by him in 2008 where he carefully described his cooperative games model, he almost expressed a personal relation with each of the variables on his slides)... – Suvrit – 2015-05-25T00:33:17.853

4@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 – 2015-05-25T08:05:37.0532On May 19th John Nash, together with Louis Nirenberg, received the Abel price; I wonder why that hasn't been mentioned so far. – Manfred Weis – 2015-05-25T11:01:22.057

A 'tragic but meaningful' life: Legendary Princeton mathematician John Nash dies http://www.princeton.edu/main/news/archive/S43/27/52G52/index.xml?section= via @Princeton

– disposed to learn – 2015-05-28T01:33:49.8631

Link from http://www.newyorker.com/news/john-cassidy/the-triumph-and-failure-of-john-nashs-game-theory points to this question....

– woliveirajr – 2015-05-28T18:08:12.680Disappointingly, the text of the link in the New Yorker article is "his work in algebraic geometry". Sigh... I suppose one of his results at least has implications in real algebraic geometry. – Paul Siegel – 2015-05-28T23:30:29.393

Instantly Nash could calculate that there are 2 accumulation points in distinct sequence of pi. – Takahiro Waki – 2016-06-26T12:35:50.040