[1][2] Nash and fellow game theorists John Harsanyi and Reinhard Selten were awarded the 1994 Nobel Prize in Economics.

This work, also introducing a preliminary form of the Nash–Moser theorem, was later recognized by the American Mathematical Society with the Leroy P. Steele Prize for Seminal Contribution to Research.

Ennio De Giorgi and Nash found, with separate methods, a body of results paving the way for a systematic understanding of elliptic and parabolic partial differential equations.

In 1959, Nash began showing clear signs of mental illness, and spent several years at psychiatric hospitals being treated for schizophrenia.

[17][18] The thesis, written under the supervision of doctoral advisor Albert W. Tucker, contained the definition and properties of the Nash equilibrium, a crucial concept in non-cooperative games.

[28] From Warren Ambrose, a differential geometer, he learned about the conjecture that any Riemannian manifold is isometric to a submanifold of Euclidean space.

Starting with work of Camillo De Lellis and László Székelyhidi, the ideas of Nash's proof were applied for various constructions of turbulent solutions of the Euler equations in fluid mechanics.

The usual formulations of the implicit function theorem are inapplicable, for technical reasons related to the loss of regularity phenomena.

Nash's resolution of this issue, given by deforming an isometric embedding by an ordinary differential equation along which extra regularity is continually injected, is regarded as a fundamentally novel technique in mathematical analysis.

[37] Nash's paper was awarded the Leroy P. Steele Prize for Seminal Contribution to Research in 1999, where his "most original idea" in the resolution of the loss of regularity issue was cited as "one of the great achievements in mathematical analysis in this century".

It has been extended and generalized by a number of other authors, among them Gromov, Richard Hamilton, Lars Hörmander, Jacob Schwartz, and Eduard Zehnder.

...  what Nash discovered in the course of his constructions of isometric embeddings is far from 'classical' – it is something that brings about a dramatic alteration of our understanding of the basic logic of analysis and differential geometry.

Judging from the classical perspective, what Nash has achieved in his papers is as impossible as the story of his life ... [H]is work on isometric immersions ... opened a new world of mathematics that stretches in front of our eyes in yet unknown directions and still waits to be explored.While spending time at the Courant Institute in New York City, Louis Nirenberg informed Nash of a well-known conjecture in the field of elliptic partial differential equations.

After extensive discussions with Nirenberg and Lars Hörmander, Nash was able to extend Morrey's results, not only to functions of more than two variables, but also to the context of parabolic partial differential equations.

[41][42][43][44] Soon after, Nash learned from Paul Garabedian, recently returned from Italy, that the then-unknown Ennio De Giorgi had found nearly identical results for elliptic partial differential equations.

De Giorgi and Moser's methods became particularly influential over the next several years, through their developments in the works of Olga Ladyzhenskaya, James Serrin, and Neil Trudinger, among others.

[3][50] Nash's psychological issues crossed into his professional life when he gave an American Mathematical Society lecture at Columbia University in early 1959.

[54] Over the next nine years, he spent intervals of time in psychiatric hospitals, where he received both antipsychotic medications and insulin shock therapy.

[58] Encouraged by his then former wife, Lardé, Nash lived at home and spent his time in the Princeton mathematics department where his eccentricities were accepted even when his mental condition was poor.

[8] For Nash, this included seeing himself as a messenger or having a special function of some kind, of having supporters and opponents and hidden schemers, along with a feeling of being persecuted and searching for signs representing divine revelation.

[64] Nash wrote in 1994: I spent times of the order of five to eight months in hospitals in New Jersey, always on an involuntary basis and always attempting a legal argument for release.

And it did happen that when I had been long enough hospitalized that I would finally renounce my delusional hypotheses and revert to thinking of myself as a human of more conventional circumstances and return to mathematical research.

In 1994, he received the Nobel Memorial Prize in Economic Sciences (along with John Harsanyi and Reinhard Selten) for his game theory work as a Princeton graduate student.

[66] Nash's later work involved ventures in advanced game theory, including partial agency, which show that, as in his early career, he preferred to select his own path and problems.

He has compared not thinking in an acceptable manner, or being "insane" and not fitting into a usual social function, to being "on strike" from an economic point of view.

[74] On May 19, 2015, a few days before his death, Nash, along with Louis Nirenberg, was awarded the 2015 Abel Prize by King Harald V of Norway at a ceremony in Oslo.

[77] In Santa Monica, California, in 1954, while in his twenties, Nash was arrested for indecent exposure in a sting operation targeting gay men.

[79] Not long after breaking up with Stier, Nash met Alicia Lardé Lopez-Harrison, a naturalized U.S. citizen from El Salvador.

John Charles Martin Nash earned a PhD in mathematics from Rutgers University and was diagnosed with schizophrenia as an adult.

A film by the same name was released in 2001, directed by Ron Howard with Russell Crowe playing Nash; it won four Academy Awards, including Best Picture.