John Forbes Nash Jr.

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John Forbes Nash (2000s)

John Forbes Nash, Jr. (born June 13, 1928 in Bluefield , West Virginia , † May 23, 2015 near Monroe Township , New Jersey ) was an American mathematician who worked particularly in the areas of game theory and differential geometry, as well as in the field of partial differential equations worked. In 1994, together with Reinhard Selten and John Harsanyi, he received the Alfred Nobel Memorial Prize for Economics for their joint achievements in the field of game theory. This made Nash one of the few mathematicians to receive this award. In 2015 he received the Abel Prize , one of the most important science prizes in the field of mathematics.

After a promising start to his mathematical career, Nash fell ill with schizophrenia at the age of thirty . Nash recovered from the disease in the early 1990s. His story became known to a wider audience in late 2001 through the award-winning feature film A Beautiful Mind .

Training and work

John F. Nash (2006)

From 1945 to 1948 Nash studied at the Carnegie Institute of Technology in Pittsburgh , where he received his bachelor's degree in 1945 and his master's degree in 1948. Originally he wanted to be an engineer like his father , but developed a great passion for mathematics . He was also interested in physics and even presented one of his theories to Albert Einstein when he began to study at Princeton in 1948, but at the end of the conversation he advised him to "study more physics".

While still in Pittsburgh, he began to be interested in the negotiation problem, the solution of which John von Neumann and Oskar Morgenstern had left open in their 1944 book Theory of Games and Economic Behavior .

Nash received his doctorate in 1950 from Princeton University with the mathematician Albert W. Tucker . The thesis entitled Non-cooperative Games advanced game theory of Morgenstern and von Neumann to the so-called Nash equilibrium ( English Nash equilibrium ). Nash proved that this equilibrium - deviating from the solutions - also exists for non- zero-sum games and for more than two players.

It is based on a set of strategies (such as price policy ) from players ( competitors in the market). One situation where neither player can benefit from changing their strategy if the other players leave their strategies unchanged is a Nash equilibrium. The importance of this work from 1950 was only recognized later in connection with the further development of game theory and in 1994 earned him the Alfred Nobel Memorial Prize for Economics . Von Neumann himself was unimpressed at a meeting with Nash at the time; he considered the result to be trivial and only mentioned it indirectly in the introduction in the new edition of his book with Morgenstern on game theory from 1953. Nash himself also rated the work more as a by-product compared to his later work. In the event that his work on game theory was not accepted, he would have already prepared another work in algebraic geometry , says Nash.

In 1952 his work on real algebraic manifolds appeared , which he himself considered to be his perfect work. The idea behind it was to approximate each manifold by an algebraic variety (which were much easier to handle and describe by polynomials), possibly by moving to spaces of much higher dimension. In this context, Nash manifolds and Nash functions are named after him.

After receiving his doctorate, Nash increasingly turned to analysis , in particular differential geometry and partial differential equations. He proved that every Riemannian manifold can be isometrically embedded in the Euclidean manifold ( Nash's embedding theorem ). The question of whether this is possible was already asked by Bernhard Riemann , and the popular opinion in the 1950s was that it was not. Nash's result was unexpected and had far-reaching consequences. A partial result of his embedding theorem was used by Jürgen Moser in 1966 in the theory of nonlinear partial differential equations and is known as the Nash and Moser theorem.

From 1950, Nash spent four years in the summer months of the Rand Corporation doing clandestine research, including Kenneth Arrow , John Milnor (who worked with Nash at Rand), and others on applications of game theory to strategic Cold War situations . From 1951 to 1953 Nash Moore was an instructor at the Massachusetts Institute of Technology and from 1953 he was an assistant professor there and from 1957 to 1959 an associate professor. In 1955 he submitted a proposal for an encryption method to the National Security Agency , but received a rejection.

In 1958 he published (parallel to Ennio De Giorgi , but independently of him) a solution to the regularity problem of partial differential equations, which David Hilbert had included in his well-known list of the largest, open problems in mathematics in 1900 (19th problem). The results became known as the De Giorgi and Nash Theorem and have far-reaching consequences for the theory of partial differential equations. Nash was on leave from MIT in 1956/57 and was nominally at the Institute for Advanced Study in Princeton, but did research at the Courant Institute in New York City, the then Mecca for research into partial differential equations, where Peter Lax , Louis Nirenberg and Lars, among others, at the time Hörmander were active.

In 1947, Nash designed the game, which is now marketed under the name " Hex ", through considerations of game theory, independently of the Dane Piet Hein a few years earlier. A prototype was built by David Gale , a friend of Nash, and the game was soon popular among mathematicians at Princeton such as John Milnor. Around 1950 he spent a lot of time in Princeton with board games (especially chess , Go , where Ralph Fox was a master, and the so-called war game ) and together with other students also developed the game So Long Sucker .

Life and sickness

By the late 1950s, Nash was widely recognized as a leading mathematician, which was reflected in an article in Forbes Magazine , and he was nominated for the Fields Medal in 1958 , particularly for his work on Hilbert's 19th problem concurrent with de Giorgi. In the final evaluation, he was third behind Klaus Roth and René Thom , who finally received the Fields Medal in 1958. He was on the verge of a full professorship at MIT when the first signs of Nash's disease became apparent in 1959. In May 1959 he was diagnosed with paranoid schizophrenia . According to the Nash biographer Sylvia Nasar , Nash was now increasingly showing anti-Semitic tendencies and prone to outbreaks of violence. Nash gave up his position at MIT and, after a short hospital stay, went to Paris and Geneva in 1959/60, where he saw himself as a global citizen and exile. In Luxembourg he said he tried to return his American citizenship.

In 1961, his wife Alicia Lardé and his mother were forced to admit Nash to a mental hospital ( Trenton State Hospital). Here he was treated with insulin shock therapy, which put him into a coma. He recovered and was able to attend a conference on game theory in 1961. In 1961/62 and 1963/64 he was again at the Institute for Advanced Study, in 1962 he visited Paris, London and Geneva again, and then returned to Princeton.

In 1964 his schizophrenia became so severe that he had to be admitted to a psychiatric clinic (the private Carrier Clinic in Belle Mead , New Jersey ) for a long time , and he was in Paris again in 1965 (at the invitation of Alexander Grothendieck ). For the next 20 years he was repeatedly in hospitals for relapses. As a result of his illness, he did not bring out any publications between 1966 and 1996. Before that, however, some outstanding works appeared. From the 1960s came an idea in the theory of the resolution of singularities in algebraic geometry known as Nash Blowing Up (so called by Heisuke Hironaka , whom Nash orally conveyed the idea to), and some influential work on partial differential equations. From 1965 to 1967 Nash was at MIT , supported by prominent US mathematicians such as John Milnor, who knew him from college days . From the 1970s to 1990s he lived in Princeton, where he could be seen regularly on campus. While he initially caught the students' attention with strange messages he left behind, mathematicians at Princeton (like Peter Sarnak ) began to notice by the early 1990s that he had regained some of his old problem-solving skills. In his last years he increasingly turned to monetary theory, advocating index money .

He was married to Alicia Lardé for the second time since 2001 (first marriage from 1957, divorced in 1963). They had a son (* 1959); he also had a son (* 1953) from a previous relationship. Nash died along with his wife in a traffic accident on the New Jersey Turnpike in May 2015 ; they were in a taxi on their way home from receiving the Abel Prize . Both were not wearing their seat belts and were thrown out of the vehicle.


Nash received honorary doctorates from the University of Athens and Carnegie Mellon University in Pittsburgh. He was elected to the American Academy of Arts and Sciences in 1995 , the National Academy of Sciences in 1996, and the American Philosophical Society in 2006, and was a Fellow of the American Mathematical Society .

Film adaptations

The 2001 feature film, A Beautiful Mind, starring Russell Crowe , tells the story of Nash's ingenious designs, illness, and recovery; the film won four Academy Awards in 2002 . The script is based on the 1998 biography of the same name by Sylvia Nasar. The film adaptation only agrees in corner points with Nash's biography; many details are fictitious.

In addition, the life of Nash has also been portrayed in documentaries:

  • A Brilliant Madness: The story of Nobel Prize winning mathematician John Nash by Mark Samels and Randall MacLowry with the participation of Sylvia Nasar. A Yellow Jersey Films production for American Experience , USA 2002. 60 min, in English.
  • John Nash: A Beautiful Genius. - An unauthorized tribute , by Sean Buckley and Guy Portner. Buck Productions Inc., Canada 2002. 53 min, in English.
  • A Mind on Strike - John Nash revisited (2017): The last years of Nobel Laureate John Nash in a film by Peter Badge and Jim Rakete, May 31, 2017, in English


  • With Edward Elgar (Ed.): Essays on Game Theory. 1996, ISBN 1-85898-426-2 .
  • The essential John Nash , edited by Harold W. Kuhn and Sylvia Nasar, Princeton University Press, 2002, ISBN 0-691-09527-2 .


  • János Kollár : Nash´s work in algebraic geometry, Bulletin AMS, 2016, online
  • John Milnor: John Nash and "A Beautiful Mind". (PDF; 116 kB) In: Notices of the AMS. November 1998.
  • Sylvia Nasar: genius and madness. The life of the brilliant mathematician John Nash. 9th edition, Piper Verlag, Munich 2005, ISBN 3-492-23674-X (Original: A beautiful mind , Simon and Schuster, 1998)
  • Tom Siegfried: A Beautiful Math: John Nash, Game Theory and the modern quest for a code of nature. Joseph Henry Press, Washington DC 2006.
  • Martin A. Nowak : John Forbes Nash (1928-2015). In: Nature . Volume 522, No. 7557, 2015, p. 420, doi: 10.1038 / 522420a
  • Martin Shubik : John Forbes Nash Jr. (1928-2015). In: Science . Volume 348, No. 6241, 2015, p. 1324, doi: 10.1126 / science.aac7085
  • Nash-dedicated issue of the Bulletin of the AMS, Volume 54, 2017, No. 2, Online (Gromov on the embedding theorem, Kollar on algebraic geometry, Klainerman on analysis, De Lellis and Szekelyhidi on the h-principle at PDE)

Web links

Commons : John Forbes Nash Jr.  album with pictures, videos and audio files

Individual evidence

  1. Sylvia Nasar: Beautiful Mind , 1998, pp. 70f
  2. ^ John Nash: The bargaining problem , Econometrica, Volume 18, 1950, pp. 155-162. Reprinted in The Essential John Nash .
  3. John Nash: Non-cooperative games , 1950, online version ( Memento from September 17, 2012 in the Internet Archive ) (PDF; 1.2 MB)
  4. ^ John Nash: Equilibrium points in n-person games , Proc. Nat. Acad. Sci., Volume 36, 1950, pp. 48-49, here online , John Nash: Non cooperative games , Annals of Mathematics, Volume 54, 1951, pp. 286-295 ( JSTOR ). Both reprinted in The Essential John Nash .
  5. Sylvia Nasar, Introduction to The Essential John Nash , p. XIX
  6. Nash in his autobiography in The Essential John Nash
  7. ^ John Nash: Real algebraic manifolds , Annals of Mathematics, Volume 56, 1952, pp. 405-421
  8. Sylvia Nasar in the introduction to The Essential John Nash , p. XXI
  9. John Nash: The imbedding problem for Riemannian Manifolds , Annals of Mathematics, Volume 63, 1956, pp. 20-63. Reprinted in The Essential John Nash .
  10. Jürgen Moser: A rapidly convergent iteration method and non-linear partial differential equations , part 1 and 2, Ann. Scuola Norm. Sup. Pisa, Vol. 20, 1966, pp. 265, 499
  11. ^ Richard Hamilton: The inverse function theorem of Nash and Moser , BAMS, 1982
  12. ^ Correspondence with the NSA ,, May 25, 2015
  13. ^ John Nash: Continuity of solutions of Parabolic and Elliptic Equations , American Journal of Mathematics, Volume 80, 1958, pp. 931-954, reprinted in The Essential John Nash . Shorter communication in Nash Parabolic equations , Proc. Nat. Acad. Sci., Vol. 53, 1957, pp. 754-758
  14. John Milnor: A Noble Prize for John Nash , Mathematical Intelligencer, Volume 17, 1995, Issue 3. Sylvia Nasar: Beautiful Mind , p 76. John Milnor, Interview Notices AMS February 2012, mentioned that it from them at that time Nash called has been.
  15. He also tried unsuccessfully to market it, including at Parker Brothers, who released it as Hex in the mid-1950s.
  16. Specifically against Norman Steenrod and John Tukey . The game was referred to in English with the German word. According to John Milnor, Interview Notices AMS March 2012, it was not a question of the simulation game variant, but the chess variant.
  17. Michael Barany, The Fields Medal should return to its roots , Nature, Volume 553, 2018, pp. 271-273
  18. Sylvia Nasar: Beautiful Mind , chapter 34
  19. Nash in his autobiography in The Essential John Nash .
  20. Patrick Bernau : People are not always rational, interview. In: Conclusion blog. 2010, accessed October 7, 2016 .
  21. The therapy was considered progressive at the time, while the previously widespread electric shock therapy was largely abandoned.
  22. Nash: Arc structure of singularities , in: Duke J. Math. , Volume 81, 1995, p. 31 (written in 1966 and then circulating as a preprint), Nash: Analyticity of solutions of implicit function problems with analytic data , in: Annals of Mathematics , Volume 84, 1966, p. 345, Nash: Le problemème de Cauchy pour les equations différentielles d'un fluide générale , in: Bull. Soc. Math. De France , Volume 90, 1962, p. 487, online ( Memento of the original from February 24, 2014 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. (written during his stay at the Trenton Clinic). @1@ 2Template: Webachiv / IABot /
  23. Keynesians love inflation
  24. Famed 'A Beautiful Mind' mathematician John Nash, wife killed in taxi crash, police say ,, May 24, 2015
  25. See also for a detailed description.
  26. See also