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Tucker]] | known_for = {{plainlist| * [[Cooperative bargaining#Nash bargaining game|Nash bargaining game]] * [[Nash blowing-up]] * [[Nash equilibrium]] * [[Nash embedding theorem]] * [[Nash functions]] * [[Sobolev inequality#Nash inequality|Nash inequality]] * [[Hilbert's nineteenth problem#Nash's theorem|Nash's theorem]] * [[Nash–Moser theorem]] * [[Hilbert's nineteenth problem]] * Ideal money}} | awards = {{plainlist| * [[John von Neumann Theory Prize]] (1978) * [[Nobel Memorial Prize in Economic Sciences]] (1994) * [[Member of the National Academy of Sciences]] (1996) * [[Abel Prize]] (2015)}} | spouse = {{ubl|{{marriage|[[Alicia Nash|Alicia Lardé Lopez-Harrison]]|1957|1963|end=div}}|{{marriage||2001|2015|end=their deaths}}}} | children = 2 }} '''John Forbes Nash, Jr.''' (June 13, 1928&nbsp;– May 23, 2015), known and published as '''John Nash''', was an American mathematician who made fundamental contributions to [[game theory]], [[real algebraic geometry]], [[differential geometry]], and [[partial differential equation]]s.<ref>{{cite news |author=Goode, Erica |title=John F. Nash Jr., Math Genius Defined by a 'Beautiful Mind,' Dies at 86 |url=https://www.nytimes.com/2015/05/25/science/john-nash-a-beautiful-mind-subject-and-nobel-winner-dies-at-86.html |newspaper=[[The New York Times]] |date=May 24, 2015}}</ref><ref>{{cite web |title=John F. Nash Jr. and Louis Nirenberg share the Abel Prize |url=http://www.abelprize.no/nyheter/vis.html?tid=63589 |publisher=[[Abel Prize]] |date=March 25, 2015 |access-date=May 27, 2015 |archive-date=June 16, 2019 |archive-url=https://web.archive.org/web/20190616214625/https://www.abelprize.no/nyheter/vis.html?tid=63589 }}</ref> Nash and fellow game theorists [[John Harsanyi]] and [[Reinhard Selten]] were awarded the 1994 [[Nobel Memorial Prize in Economic Sciences|Nobel Prize in Economics]]. In 2015, he and [[Louis Nirenberg]] were awarded the [[Abel Prize]] for their contributions to the field of partial differential equations. As a graduate student in the [[Princeton University Department of Mathematics]], Nash introduced a number of concepts (including [[Nash equilibrium]] and the [[Nash bargaining solution]]) which are now considered central to game theory and its applications in various sciences. In the 1950s, Nash discovered and proved the [[Nash embedding theorems]] by solving a system of nonlinear partial differential equations arising in [[Riemannian geometry]]. 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 partial differential equation|elliptic]] and [[parabolic partial differential equation]]s. Their De Giorgi–Nash theorem on the smoothness of solutions of such equations resolved [[Hilbert's nineteenth problem]] on regularity in the [[calculus of variations]], which had been a well-known [[open problem]] for almost sixty years. In 1959, Nash began showing clear signs of mental illness, and spent several years at [[psychiatric hospital]]s being treated for [[schizophrenia]]. After 1970, his condition slowly improved, allowing him to return to academic work by the mid-1980s.<ref name="Nasar1994" /> Nash's life was the subject of [[Sylvia Nasar]]'s 1998 biographical book ''[[A Beautiful Mind (book)|A Beautiful Mind]]'', and his struggles with his illness and his recovery became the basis for a [[A Beautiful Mind (film)|film of the same name]] directed by [[Ron Howard]], in which Nash was portrayed by [[Russell Crowe]].<ref name="Oscar race scrutinizes">{{cite news |title=Oscar race scrutinizes movies based on true stories |work=[[USA Today]] |date=March 6, 2002 |url=http://usatoday30.usatoday.com/life/movies/oscar2002/2002-03-06-true-stories.htm |access-date=January 22, 2008}}</ref><ref>{{cite news |url=http://usatoday30.usatoday.com/life/movies/oscar2002/2002-03-24-winners.htm |title=Academy Award Winners |work=[[USA Today]] |access-date=August 30, 2008 |date=March 25, 2002}}</ref><ref>{{cite web |last=Yuhas |first=Daisy |title=Throughout History, Defining Schizophrenia Has Remained A Challenge (Timeline) |url=http://www.scientificamerican.com/article/throughout-history-defining-schizophrenia-has-remained-challenge/ |work=[[Scientific American Mind]] |date=March 2013 |access-date=March 2, 2013}}</ref> == Early life and education == John Forbes Nash Jr. was born on June 13, 1928, in [[Bluefield, West Virginia]]. His father and namesake, John Forbes Nash Sr., was an [[electrical engineer]] for the [[American Electric Power#Appalachian Power|Appalachian Electric Power Company]]. His mother, Margaret Virginia (née Martin) Nash, had been a schoolteacher before she was married. He was baptized in the [[Episcopal Church (United States)|Episcopal Church]].{{sfnm|1a1=Nasar|1y=1998|1loc=Chapter 1}} He had a younger sister, Martha (born November 16, 1930).<ref name="Nash1995">{{cite conference|last1=Nash|first1=John F. Jr.|year=1995|url=https://www.nobelprize.org/nobel_prizes/economic-sciences/laureates/1994/nash-bio.html|title=John F. Nash Jr. – Biographical|book-title=The Nobel Prizes 1994: Presentations, Biographies & Lectures|editor-first1=Tore|editor-last1=Frängsmyr|publisher=[[Nobel Foundation]]|location=Stockholm|editor-link1=Tore Frängsmyr|isbn=978-91-85848-24-9|pages=275–279}}</ref> Nash attended kindergarten and public school, and he learned from books provided by his parents and grandparents.<ref name="Nash1995" /> Nash's parents pursued opportunities to supplement their son's education, and arranged for him to take advanced mathematics courses at nearby Bluefield College (now [[Bluefield University]]) during his final year of high school. He attended [[Carnegie Mellon University|Carnegie Institute of Technology]] (which later became Carnegie Mellon University) through a full benefit of the George Westinghouse Scholarship, initially majoring in [[chemical engineering]]. He switched to a [[chemistry]] major and eventually, at the advice of his teacher [[John Lighton Synge]], to mathematics. After graduating in 1948, with both a [[Bachelor of Science|B.S.]] and [[Master of Science|M.S.]] in mathematics, Nash accepted a fellowship to [[Princeton University]], where he pursued further [[Princeton University Department of Mathematics|graduate studies]] in mathematics and sciences.<ref name="Nash1995" /> Nash's adviser and former Carnegie professor [[Richard Duffin]] wrote a letter of recommendation for Nash's entrance to Princeton stating, "He is a mathematical genius."<ref>{{cite web |title=Nash recommendation letter |url=https://webspace.princeton.edu/users/mudd/Digitization/AC105/AC105_Nash_John_Forbes_1950.pdf |archive-url=https://web.archive.org/web/20170607041209/https://webspace.princeton.edu/users/mudd/Digitization/AC105/AC105_Nash_John_Forbes_1950.pdf |archive-date=June 7, 2017 |access-date=June 5, 2015 |page=23}}</ref><ref>{{cite web |editor1-first=Harold W. |editor1-last=Kuhn |editor2-first=Sylvia |editor2-last=Nasar |editor-link=Sylvia Nasar |url=http://press.princeton.edu/chapters/i7238.pdf |archive-url=https://web.archive.org/web/20070101170703/http://press.princeton.edu/chapters/i7238.pdf |archive-date=2007-01-01 |url-status=live |title=The Essential John Nash |publisher=[[Princeton University Press]] |pages=Introduction, xi |access-date=April 17, 2008}}</ref> Nash was also accepted at [[Harvard University]]. However, the chairman of the mathematics department at Princeton, [[Solomon Lefschetz]], offered him the [[John Stewart Kennedy|John S. Kennedy]] fellowship, convincing Nash that Princeton valued him more.{{sfnm|1a1=Nasar|1y=1998|1loc=Chapter 2}} Further, he considered Princeton more favorably because of its proximity to his family in Bluefield.<ref name="Nash1995" /> At Princeton, he began work on his equilibrium theory, later known as the [[Nash equilibrium]].<ref>Nasar (2002), pp. xvi–xix.</ref> == Research contributions == [[File:John f nash 20061102 3.jpg|thumb|upright|Nash in November 2006 at a [[game theory]] conference in [[Cologne]], Germany]] Nash did not publish extensively, although many of his papers are considered landmarks in their fields.<ref>{{cite journal | first = John | last = Milnor | author-link = John Milnor |year = 1998 | title = John Nash and 'A Beautiful Mind' | url = https://www.ams.org/notices/199810/milnor.pdf| journal = [[Notices of the American Mathematical Society]] | volume = 25 | issue = 10|pages = 1329–1332 }}</ref> As a graduate student at Princeton, he made foundational contributions to [[game theory]] and [[real algebraic geometry]]. As a postdoctoral fellow at [[MIT]], Nash turned to [[differential geometry]]. Although the results of Nash's work on differential geometry are phrased in a geometrical language, the work is almost entirely to do with the [[mathematical analysis]] of [[partial differential equations]].<ref name="steele">{{cite journal|title=1999 Steele Prizes|journal=[[Notices of the American Mathematical Society]]|date=April 1999|pages=457–462|volume=46|issue=4|url=https://www.ams.org/notices/199904/comm-steele-prz.pdf |archive-url=https://web.archive.org/web/20000829221437/http://www.ams.org/notices/199904/comm-steele-prz.pdf |archive-date=2000-08-29 |url-status=live}}</ref> After proving his two [[Nash embedding theorems|isometric embedding theorem]]s, Nash turned to research dealing directly with partial differential equations, where he discovered and proved the De Giorgi–Nash theorem, thereby resolving one form of [[Hilbert's nineteenth problem]]. In 2011, the [[National Security Agency]] declassified letters written by Nash in the 1950s, in which he had proposed a new [[encryption]]–decryption machine.<ref>{{cite web |url=https://www.nsa.gov/Press-Room/Press-Releases-Statements/Press-Release-View/Article/1630570/national-cryptologic-museum-opens-new-exhibit-on-dr-john-nash/ |publisher=[[National Security Agency]] |title=2012 Press Release – National Cryptologic Museum Opens New Exhibit on Dr. John Nash |access-date=July 30, 2022}}</ref> The letters show that Nash had anticipated many concepts of modern [[cryptography]], which are based on [[Computational hardness assumption|computational hardness]].<ref>{{cite web |url=http://agtb.wordpress.com/2012/02/17/john-nashs-letter-to-the-nsa/ |title=John Nash's Letter to the NSA; Turing's Invisible Hand |access-date=February 25, 2012|date=February 17, 2012 }}</ref> === Game theory === Nash earned a PhD in 1950 with a 28-page dissertation on [[Non-cooperative game theory|non-cooperative games]].<ref name="JohnNash_PhD">{{cite web |last1=Nash |first1=John F. |author-link=John Forbes Nash Jr. |title=Non-Cooperative Games |work=PhD thesis |publisher=Princeton University |date=May 1950 |url=https://www.princeton.edu/mudd/news/faq/topics/Non-Cooperative_Games_Nash.pdf |access-date=May 24, 2015 |archive-url=https://web.archive.org/web/20150420144847/http://www.princeton.edu/mudd/news/faq/topics/Non-Cooperative_Games_Nash.pdf |archive-date=April 20, 2015 }}</ref><ref>{{cite book |first=Martin J. |last=Osborne |date=2004 |title=An Introduction to Game Theory |url=https://archive.org/details/introductiontoga00osbo |url-access=limited |publisher=[[Oxford University Press]] |location=Oxford, England |page=[https://archive.org/details/introductiontoga00osbo/page/n34 23] |isbn=0-19-512895-8}}</ref> 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. A version of his thesis was published a year later in the [[Annals of Mathematics]].{{sfnm|1a1=Nash|1y=1951}} In the early 1950s, Nash carried out research on a number of related concepts in game theory, including the theory of [[cooperative game theory|cooperative games]].{{sfnm|1a1=Nash|1y=1950a|2a1=Nash|2y=1950b|3a1=Nash|3y=1953}} For his work, Nash was one of the recipients of the [[Nobel Memorial Prize in Economic Sciences]] in 1994. === Real algebraic geometry === In 1949, while still a graduate student, Nash found a new result in the mathematical field of [[real algebraic geometry]].{{sfnm|1a1=Nasar|1y=1998|1loc=Chapter 15}} He announced his theorem in a contributed paper at the [[International Congress of Mathematicians]] in 1950, although he had not yet worked out the details of its proof.{{sfnm|1a1=Nash|1y=1952a}} Nash's theorem was finalized by October 1951, when Nash submitted his work to the [[Annals of Mathematics]].{{sfnm|1a1=Nash|1y=1952b}} It had been well-known since the 1930s that every [[closed manifold|closed]] [[smooth manifold]] is [[diffeomorphic]] to the [[zero set]] of some collection of [[smooth function]]s on [[Euclidean space]]. In his work, Nash proved that those smooth functions can be taken to be [[polynomial]]s.<ref name="bochnak">{{cite book|mr=1659509|last1=Bochnak|first1=Jacek|last2=Coste|first2=Michel|last3=Roy|first3=Marie-Françoise|title=Real algebraic geometry|edition=Translated and revised from 1987 French original|series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete|Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge]]|volume=36|publisher=[[Springer-Verlag]]|location=Berlin|year=1998|isbn=3-540-64663-9|zbl=0912.14023|doi=10.1007/978-3-662-03718-8|s2cid=118839789 |author-link3=Marie-Françoise Roy}}</ref> This was widely regarded as a surprising result,{{sfnm|1a1=Nasar|1y=1998|1loc=Chapter 15}} since the class of smooth functions and smooth manifolds is usually far more flexible than the class of polynomials. Nash's proof introduced the concepts now known as [[Nash function]] and [[Nash manifold]], which have since been widely studied in real algebraic geometry.<ref name="bochnak" /><ref>{{cite book|mr=0904479|last1=Shiota|first1=Masahiro|title=Nash Manifolds |series=[[Lecture Notes in Mathematics]]|volume=1269|publisher=[[Springer-Verlag]]|location=Berlin|year=1987|isbn=3-540-18102-4|doi=10.1007/BFb0078571|zbl=0629.58002}}</ref> Nash's theorem itself was famously applied by [[Michael Artin]] and [[Barry Mazur]] to the study of [[dynamical system]]s, by combining Nash's polynomial approximation together with [[Bézout's theorem]].<ref>{{cite journal|mr=0176482|last1=Artin|first1=M.|last2=Mazur|first2=B.|title=On periodic points|journal=[[Annals of Mathematics]]|series=Second Series|volume=81|year=1965|pages=82–99|issue=1|doi=10.2307/1970384|jstor=1970384 |zbl=0127.13401|author-link1=Michael Artin|author-link2=Barry Mazur}}</ref><ref>{{cite journal|first1=Mikhaïl|last1=Gromov|url=https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/1024.pdf|title=On the entropy of holomorphic maps|journal=[[L'Enseignement mathématique|L'Enseignement Mathématique. Revue Internationale]]|series=2e Série|volume=49|year=2003|issue=3–4|pages=217–235|mr=2026895|zbl=1080.37051|author-link1=Mikhael Gromov (mathematician)}}</ref> ===Differential geometry=== During his postdoctoral position at [[MIT]], Nash was eager to find high-profile mathematical problems to study.{{sfnm|1a1=Nasar|1y=1998|1loc=Chapter 20}} From [[Warren Ambrose]], a [[differential geometry|differential geometer]], he learned about the conjecture that any [[Riemannian manifold]] is [[isometry|isometric]] to a [[submanifold]] of [[Euclidean space]]. Nash's results proving the conjecture are now known as the [[Nash embedding theorem]]s, the second of which [[Mikhael Gromov (mathematician)|Mikhael Gromov]] has called "one of the main achievements of mathematics of the twentieth century".<ref name="Nash2015">{{cite conference|mr=3470099|book-title=Open problems in mathematics|editor-first1=John Forbes Jr.|editor-last1=Nash|editor-first2=Michael Th.|editor-last2=Rassias|publisher=[[Springer, Cham]]|year=2016|isbn=978-3-319-32160-8|doi=10.1007/978-3-319-32162-2|first1=Misha|last1=Gromov|author-link1=Mikhael Gromov (mathematician)|title=Introduction John Nash: theorems and ideas|arxiv=1506.05408}}</ref> Nash's first embedding theorem was found in 1953.{{sfnm|1a1=Nasar|1y=1998|1loc=Chapter 20}} He found that any Riemannian manifold can be isometrically embedded in a Euclidean space by a [[continuously differentiable]] mapping.{{sfnm|1a1=Nash|1y=1954}} Nash's construction allows the [[codimension]] of the embedding to be very small, with the effect that in many cases it is logically impossible that a highly-differentiable isometric embedding exists. (Based on Nash's techniques, [[Nicolaas Kuiper]] soon found even smaller codimensions, with the improved result often known as the ''Nash–Kuiper theorem''.) As such, Nash's embeddings are limited to the setting of low differentiability. For this reason, Nash's result is somewhat outside the mainstream in the field of [[differential geometry]], where high differentiability is significant in much of the usual analysis.<ref>{{cite book|last1=Eliashberg|first1=Y.|last2=Mishachev|first2=N.|title=Introduction to the h-principle|series=[[Graduate Studies in Mathematics]]|volume=48|publisher=[[American Mathematical Society]]|location=Providence, RI|year=2002|isbn=0-8218-3227-1|mr=1909245|author-link1=Yakov Eliashberg|doi=10.1090/gsm/048}}</ref><ref name="pdr">{{cite book|last1=Gromov|first1=Mikhael|title=Partial differential relations|series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3)|volume=9|publisher=[[Springer-Verlag]]|location=Berlin|year=1986|isbn=3-540-12177-3|mr=0864505|author-link1=Mikhael Gromov (mathematician)|doi=10.1007/978-3-662-02267-2}}</ref> However, the logic of Nash's work has been found to be useful in many other contexts in [[mathematical analysis]]. 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]].<ref>{{cite journal|last1=De Lellis|first1=Camillo|last2=Székelyhidi|first2=László Jr.|title=Dissipative continuous Euler flows|journal=[[Inventiones Mathematicae]]|volume=193|year=2013|issue=2|pages=377–407|mr=3090182|author-link1=Camillo De Lellis|doi=10.1007/s00222-012-0429-9| arxiv=1202.1751 | bibcode=2013InMat.193..377D | s2cid=2693636 }}</ref><ref>{{cite journal|last1=Isett|first1=Philip|title=A proof of Onsager's conjecture|journal=[[Annals of Mathematics]]|series=Second Series|year=2018|volume=188|issue=3|pages=871–963|mr=3866888|doi=10.4007/annals.2018.188.3.4|s2cid=119267892|url=https://authors.library.caltech.edu/87369/|arxiv=1608.08301|access-date=October 11, 2022|archive-date=October 11, 2022|archive-url=https://web.archive.org/web/20221011050610/https://authors.library.caltech.edu/87369/|url-status=dead}}</ref> In the 1970s, [[Mikhael Gromov (mathematician)|Mikhael Gromov]] developed Nash's ideas into the general framework of ''convex integration'',<ref name="pdr" /> which has been (among other uses) applied by [[Stefan Müller (mathematician)|Stefan Müller]] and [[Vladimír Šverák]] to construct counterexamples to generalized forms of [[Hilbert's nineteenth problem]] in the [[calculus of variations]].<ref>{{cite journal|last1=Müller|first1=S.|last2=Šverák|first2=V.|title=Convex integration for Lipschitz mappings and counterexamples to regularity|journal=[[Annals of Mathematics]]|series=Second Series|volume=157|year=2003|issue=3|pages=715–742|mr=1983780|author-link1=Stefan Müller (mathematician)|author-link2=Vladimir Šverák|doi=10.4007/annals.2003.157.715| s2cid=55855605 |doi-access=free|arxiv=math/0402287}}</ref> Nash found the construction of smoothly differentiable isometric embeddings to be unexpectedly difficult.{{sfnm|1a1=Nasar|1y=1998|1loc=Chapter 20}} However, after around a year and a half of intensive work, his efforts succeeded, thereby proving the second Nash embedding theorem.{{sfnm|1a1=Nash|1y=1956}} The ideas involved in proving this second theorem are largely separate from those used in proving the first. The fundamental aspect of the proof is an [[implicit function theorem]] for isometric embeddings. 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]].<ref name="hamilton82">{{cite journal|first=Richard S.|last=Hamilton|mr=0656198|title=The inverse function theorem of Nash and Moser|journal=[[Bulletin of the American Mathematical Society]] |series=New Series |volume=7|year=1982|issue=1|pages=65–222|doi-access=free|doi=10.1090/s0273-0979-1982-15004-2|zbl=0499.58003|author-link1=Richard S. Hamilton}}</ref> 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".<ref name="steele" /> According to Gromov:<ref name="Nash2015" /> {{blockquote|You must be a novice in analysis or a genius like Nash to believe anything like that can be ever true and/or to have a single nontrivial application.}} Due to [[Jürgen Moser]]'s extension of Nash's ideas for application to other problems (notably in [[celestial mechanics]]), the resulting implicit function theorem is known as the [[Nash–Moser theorem]]. It has been extended and generalized by a number of other authors, among them Gromov, [[Richard S. Hamilton|Richard Hamilton]], [[Lars Hörmander]], [[Jacob T. Schwartz|Jacob Schwartz]], and [[Eduard Zehnder]].<ref name="pdr" /><ref name="hamilton82" /> Nash himself analyzed the problem in the context of [[analytic function]]s.{{sfnm|1a1=Nash|1y=1966}} Schwartz later commented that Nash's ideas were "not just novel, but very mysterious," and that it was very hard to "get to the bottom of it."{{sfnm|1a1=Nasar|1y=1998|1loc=Chapter 20}} According to Gromov:<ref name="Nash2015" /> {{blockquote|Nash was solving classical mathematical problems, difficult problems, something that nobody else was able to do, not even to imagine how to do it. ...&nbsp; what Nash discovered in the course of his constructions of isometric embeddings is far from 'classical'&nbsp;– 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&nbsp;... [H]is work on isometric immersions&nbsp;... opened a new world of mathematics that stretches in front of our eyes in yet unknown directions and still waits to be explored.}} ===Partial differential equations=== 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 equation]]s.{{sfnm|1a1=Nasar|1y=1998|1loc=Chapter 30}} In 1938, [[Charles Morrey]] had proved a fundamental [[elliptic regularity]] result for functions of two independent variables, but analogous results for functions of more than two variables had proved elusive. 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 equation]]s.{{sfnm|1a1=Nash|1y=1957|2a1=Nash|2y=1958}} In his work, as in Morrey's, uniform control over the continuity of the solutions to such equations is achieved, without assuming any level of differentiability on the coefficients of the equation. The [[Sobolev inequality|Nash inequality]] was a particular result found in the course of his work (the proof of which Nash attributed to [[Elias Stein]]), which has been found useful in other contexts.<ref name="davies">{{cite book|mr=0990239|last1=Davies|first1=E. B.|title=Heat kernels and spectral theory|series=Cambridge Tracts in Mathematics|volume=92|publisher=[[Cambridge University Press]]|location=Cambridge|year=1989|isbn=0-521-36136-2|doi=10.1017/CBO9780511566158|author-link1=E. Brian Davies}}</ref><ref>{{cite book|mr=2569498|last1=Grigor'yan|first1=Alexander|title=Heat kernel and analysis on manifolds|series=AMS/IP Studies in Advanced Mathematics|volume=47|publisher=[[American Mathematical Society]]|location=Providence, RI|year=2009|isbn=978-0-8218-4935-4|doi=10.1090/amsip/047}}</ref><ref>{{cite book|mr=1840042|last1=Kigami|first1=Jun|title=Analysis on fractals|series=Cambridge Tracts in Mathematics|volume=143|publisher=[[Cambridge University Press]]|location=Cambridge|year=2001|isbn=0-521-79321-1}}</ref><ref>{{cite book|mr=1817225 |last1=Lieb|first1=Elliott H.|last2=Loss|first2=Michael|title=Analysis|edition=Second edition of 1997 original|series=[[Graduate Studies in Mathematics]]|volume=14|publisher=[[American Mathematical Society]]|location=Providence, RI|year=2001|isbn=0-8218-2783-9|author-link1=Elliott Lieb|author-link2=Michael Loss}}</ref> 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.{{sfnm|1a1=Nasar|1y=1998|1loc=Chapter 30}} De Giorgi and Nash's methods had little to do with one another, although Nash's were somewhat more powerful in applying to both elliptic and parabolic equations. A few years later, inspired by De Giorgi's method, [[Jürgen Moser]] found a different approach to the same results, and the resulting body of work is now known as the De Giorgi–Nash theorem or the De Giorgi–Nash–Moser theory (which is distinct from the [[Nash–Moser theorem]]). 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.<ref>{{cite book|mr=1814364|last1=Gilbarg|first1=David|last2=Trudinger|first2=Neil S.|title=Elliptic partial differential equations of second order|edition=Reprint of the second|series=Classics in Mathematics|publisher=[[Springer-Verlag]]|location=Berlin|year=2001|isbn=3-540-41160-7|doi=10.1007/978-3-642-61798-0|author-link1=David Gilbarg|author-link2=Neil Trudinger}}</ref><ref>{{cite book|mr=1465184|last1=Lieberman|first1=Gary M.|title=Second order parabolic differential equations|publisher=[[World Scientific|World Scientific Publishing Co., Inc.]]|location=River Edge, NJ|year=1996|isbn=981-02-2883-X|doi=10.1142/3302}}</ref> Their work, based primarily on the judicious choice of [[test function]]s in the [[weak solution|weak formulation]] of partial differential equations, is in strong contrast to Nash's work, which is based on analysis of the [[heat kernel]]. Nash's approach to the De Giorgi–Nash theory was later revisited by [[Eugene Fabes]] and [[Daniel Stroock]], initiating the re-derivation and extension of the results originally obtained from De Giorgi and Moser's techniques.<ref name="davies" /><ref>{{cite journal|mr=0855753|last1=Fabes|first1=E. B.|last2=Stroock|first2=D. W.|title=A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash|journal=[[Archive for Rational Mechanics and Analysis]]|volume=96|year=1986|issue=4|pages=327–338|doi=10.1007/BF00251802 |bibcode=1986ArRMA..96..327F |s2cid=189774501 |author-link2=Daniel Stroock}}</ref> From the fact that minimizers to many functionals in the [[calculus of variations]] solve elliptic partial differential equations, [[Hilbert's nineteenth problem]] (on the smoothness of these minimizers), conjectured almost sixty years prior, was directly amenable to the De Giorgi–Nash theory. Nash received instant recognition for his work, with [[Peter Lax]] describing it as a "stroke of genius".{{sfnm|1a1=Nasar|1y=1998|1loc=Chapter 30}} Nash would later speculate that had it not been for De Giorgi's simultaneous discovery, he would have been a recipient of the prestigious [[Fields Medal]] in 1958.<ref name="Nash1995" /> Although the medal committee's reasoning is not fully known, and was not purely based on questions of mathematical merit,{{sfnm|1a1=Nasar|1y=1998|1loc=Chapter 31}} archival research has shown that Nash placed third in the committee's vote for the medal, after the two mathematicians ([[Klaus Roth]] and [[René Thom]]) who were awarded the medal that year.<ref>{{cite journal|last1=Barany|first1=Michael|title=The Fields Medal should return to its roots|journal=[[Nature (journal)|Nature]]|volume=553|date=January 18, 2018|issue=7688 |pages=271–273|doi=10.1038/d41586-018-00513-8|bibcode=2018Natur.553..271B |doi-access=free}}</ref> == Mental illness == Although Nash's [[Mental disorder|mental illness]] first began to manifest in the form of [[paranoia]], his wife later described his behavior as erratic. Nash thought that all men who wore red ties were part of a [[Communism|communist]] conspiracy against him. He mailed letters to embassies in Washington, D.C., declaring that they were establishing a government.<ref name="Nasar1994" /><ref>[[#CITEREFNasar1998|Nasar (2011)]], p. 251.</ref> Nash's psychological issues crossed into his professional life when he gave an [[American Mathematical Society]] lecture at [[Columbia University]] in early 1959. Originally intended to present proof of the [[Riemann hypothesis]], the lecture was incomprehensible. Colleagues in the audience immediately realized that something was wrong.<ref>{{cite book |first=Karl |last=Sabbagh |title=Dr. Riemann's Zeros |publisher=[[Atlantic Books]] |location=London, England |date=2003 |isbn=1-84354-100-9 |pages=[https://archive.org/details/drriemannszeros0000sabb/page/87 87–88] |url=https://archive.org/details/drriemannszeros0000sabb/page/87 }}</ref> In April 1959, Nash was admitted to [[McLean Hospital]] for one month. Based on his paranoid, persecutory [[delusions]], [[hallucinations]], and increasing [[asociality]], he was diagnosed with [[schizophrenia]].<ref>{{cite web| url=http://www.brown.edu/Courses/BI_278/Other/Clerkship/Didactics/Readings/Schizophrenia.pdf | title=Brown University Didactic Readings: DSM-IV Schizophrenia (DSM-IV-TR #295.1–295.3, 295.90) | publisher=[[Brown University]] | location=Providence, Rhode Island|access-date=June 1, 2015 | pages=1–11}}</ref><ref name="Nasar ABM">[[#CITEREFNasar1998|Nasar (2011)]], p. 32.</ref> In 1961, Nash was admitted to the [[Trenton Psychiatric Hospital|New Jersey State Hospital at Trenton]].<ref>{{MacTutor|id=Nash}}</ref> Over the next nine years, he spent intervals of time in [[psychiatric hospital]]s, where he received both [[antipsychotic]] [[pharmaceutical drug|medications]] and [[insulin shock therapy]].<ref name="Nasar ABM" /><ref name="Roger Ebert's Movie">{{cite book |first=Roger |last=Ebert|author-link=Roger Ebert|title=Roger Ebert's Movie Yearbook 2003 |publisher=[[Andrews McMeel Publishing]] |date=2002 |url=https://archive.org/details/rogerebertsmovie00roge_6 |url-access=registration |access-date=July 10, 2008 |isbn=978-0-7407-2691-0}}</ref> Although he sometimes took prescribed medication, Nash later wrote that he did so only under pressure. According to Nash, the film ''A Beautiful Mind'' inaccurately implied he was taking [[atypical antipsychotic]]s. He attributed the depiction to the screenwriter who was worried about the film encouraging people with mental illness to stop taking their medication.<ref>{{cite web|first=Marika|last=Greihsel|url=https://www.nobelprize.org/nobel_prizes/economic-sciences/laureates/1994/nash-interview.html|title=John F. Nash Jr. – Interview|website=Nobel Foundation|date=September 1, 2004|access-date=November 3, 2018}}</ref> Nash did not take any medication after 1970, nor was he committed to a hospital ever again.<ref>{{cite web|first=John Forbes|last=Nash|url=https://www.pbs.org/wgbh/amex/nash/sfeature/sf_nash_11.html|title=PBS Interview: Medication|publisher=[[PBS]]|year=2002|access-date=September 1, 2017|archive-date=June 4, 2016|archive-url=https://web.archive.org/web/20160604221411/http://www.pbs.org/wgbh/amex/nash/sfeature/sf_nash_11.html}}</ref> Nash recovered gradually.<ref>Nash, John [https://www.pbs.org/wgbh/amex/nash/sfeature/sf_nash_14.html "PBS Interview: How does Recovery Happen?"] {{Webarchive|url=https://web.archive.org/web/20160606080035/http://www.pbs.org/wgbh/amex/nash/sfeature/sf_nash_14.html |date=June 6, 2016 }} 2002.</ref> 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. Lardé credits his [[Recovery approach|recovery]] to maintaining "a quiet life" with [[social support]].<ref name="Nasar1994" /> Nash dated the start of what he termed "mental disturbances" to the early months of 1959, when his wife was pregnant. He described a process of change "from scientific rationality of thinking into the delusional thinking characteristic of persons who are psychiatrically diagnosed as 'schizophrenic' or 'paranoid schizophrenic{{'"}}.<ref name="Nash1995" /> 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.<ref>Nash, John [https://www.pbs.org/wgbh//amex/nash/sfeature/sf_nash_12.html "PBS Interview: Delusional Thinking"] {{Webarchive|url=https://web.archive.org/web/20161001215421/http://www.pbs.org/wgbh/amex/nash/sfeature/sf_nash_12.html |date=October 1, 2016 }}. 2002.</ref> During his psychotic phase, Nash also [[Illeism|referred to himself in the third person]] as "Johann von Nassau".{{sfnm|1a1=Nasar|1y=1998|1loc=Chapter 39}} Nash suggested his delusional thinking was related to his unhappiness, his desire to be recognized, and his characteristic way of thinking, saying, "I wouldn't have had good scientific ideas if I had thought more normally." He also said, "If I felt completely pressureless I don't think I would have gone in this pattern".<ref>Nash, John [https://www.pbs.org/wgbh/amex/nash/sfeature/sf_nash_05.html "PBS Interview: The Downward Spiral"] {{Webarchive|url=https://web.archive.org/web/20170310042743/http://www.pbs.org/wgbh/amex/nash/sfeature/sf_nash_05.html |date=March 10, 2017 }} 2002.</ref> Nash reported that he started hearing voices in 1964, then later engaged in a process of consciously rejecting them.<ref>Nash, John [https://www.pbs.org/wgbh/amex/nash/sfeature/sf_nash_06.html "PBS Interview: Hearing voices"] {{Webarchive|url=https://web.archive.org/web/20120309213637/http://www.pbs.org/wgbh/amex/nash/sfeature/sf_nash_06.html |date=March 9, 2012 }}. 2002.</ref> He only renounced his "dream-like delusional hypotheses" after a prolonged period of involuntary commitment in mental hospitals—"enforced rationality". Upon doing so, he was temporarily able to return to productive work as a mathematician. By the late 1960s, he relapsed.<ref>Nash, John [https://www.pbs.org/wgbh/amex/nash/sfeature/sf_nash_13.html "PBS Interview: Paths to Recovery"] {{Webarchive|url=https://web.archive.org/web/20160605170425/http://www.pbs.org/wgbh/amex/nash/sfeature/sf_nash_13.html |date=June 5, 2016 }}. 2002.</ref> Eventually, he "intellectually rejected" his "{{not a typo|delusionally}} influenced" and "politically oriented" thinking as a waste of effort.<ref name="Nash1995" /> In 1995, he said that he did not realize his full potential due to nearly 30 years of mental illness.<ref name=Experiences>Nash, John [https://www.pbs.org/wgbh/amex/nash/sfeature/sf_nash_08.html "John Nash: My experience with mental illness"] {{Webarchive|url=https://web.archive.org/web/20161207174723/http://www.pbs.org/wgbh/amex/nash/sfeature/sf_nash_08.html |date=December 7, 2016 }}. PBS Interview, 2002.</ref> Nash wrote in 1994: {{blockquote|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 these interludes of, as it were, enforced rationality, I did succeed in doing some respectable mathematical research. Thus there came about the research for "Le problème de Cauchy pour les équations différentielles d'un fluide général"; the idea that Prof. [[Heisuke Hironaka]] called "the Nash blowing-up transformation"; and those of "Arc Structure of Singularities" and "Analyticity of Solutions of Implicit Function Problems with Analytic Data". But after my return to the dream-like delusional hypotheses in the later 60s I became a person of {{not a typo|delusionally}} influenced thinking but of relatively moderate behavior and thus tended to avoid hospitalization and the direct attention of psychiatrists. Thus further time passed. Then gradually I began to intellectually reject some of the {{not a typo|delusionally}} influenced lines of thinking which had been characteristic of my orientation. This began, most recognizably, with the rejection of politically oriented thinking as essentially a hopeless waste of intellectual effort. So at the present time I seem to be thinking rationally again in the style that is characteristic of scientists.<ref name="Nash1995" />}} == Recognition and later career == [[File:John Forbes Nash, Jr..jpg|thumb|right|Nash pictured in 2011]] In 1978, Nash was awarded the [[John von Neumann Theory Prize]] for his discovery of non-cooperative equilibria, now called Nash Equilibria. He won the [[Leroy P. Steele Prize]] in 1999. 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.<ref>Nasar (2002), p. xiii.</ref> In the late 1980s, Nash had begun to use email to gradually link with working mathematicians who realized that he was {{em|the}} John Nash and that his new work had value. They formed part of the nucleus of a group that contacted the [[Bank of Sweden]]'s Nobel award committee and were able to vouch for Nash's mental health and ability to receive the award.<ref>{{cite journal|title=The Work of John Nash in Game Theory|journal=Nobel Seminar|date=December 8, 1994|url=https://www.nobelprize.org/nobel_prizes/economic-sciences/laureates/1994/nash-lecture.pdf|archive-url=https://web.archive.org/web/20130810134711/https://www.nobelprize.org/nobel_prizes/economic-sciences/laureates/1994/nash-lecture.pdf|archive-date=August 10, 2013|access-date=May 29, 2015}}</ref> 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. Between 1945 and 1996, he published 23 scientific papers. Nash has suggested hypotheses on mental illness. He has compared not thinking in an acceptable manner, or being "insane" and not fitting into a usual social function, to being "on [[Strike action|strike]]" from an economic point of view. He advanced views in [[evolutionary psychology]] about the potential benefits of apparently nonstandard behaviors or roles.<ref>{{cite web |last=Neubauer |first=David |date=June 1, 2007 |url=http://health.yahoo.com/experts/depression/8207/john-nash-and-a-beautiful-mind-on-strike/ |title=John Nash and a Beautiful Mind on Strike |website=Yahoo! Health |archive-url=https://web.archive.org/web/20080421192944/http://health.yahoo.com/experts/depression/8207/john-nash-and-a-beautiful-mind-on-strike/ |archive-date=April 21, 2008}}</ref> Nash criticized [[Keynesian economics|Keynesian ideas]] of [[monetary economics]] which allowed for a [[central bank]] to implement [[Monetary policy|monetary policies]].<ref name=":0" /> He proposed a standard of "Ideal Money" pegged to an "industrial consumption [[price index]]" which was more stable than "bad money." He noted that his thinking on money and the function of [[Ministry of finance|monetary authority]] paralleled that of economist [[Friedrich Hayek]].{{sfnm|1a1=Nash|1y=2002a}}<ref name=":0">Zuckerman, Julia (April 27, 2005) [http://www.browndailyherald.com/2005/04/27/nobel-winner-nash-critiques-economic-theory/ "Nobel winner Nash critiques economic theory"]. ''The Brown Daily Herald''. By JULIA ZUCKERMAN Wednesday, April 27, 2005</ref> Nash received an honorary degree, Doctor of Science and Technology, from [[Carnegie Mellon University]] in 1999, an honorary degree in economics from the [[University of Naples Federico II]] in 2003,<ref>{{cite web |url=http://ricerca.repubblica.it/repubblica/archivio/repubblica/2003/03/19/napoli-laurea-nash-il-genio-dei-numeri.html |title=Napoli, laurea a Nash il 'genio dei numeri' |date=March 19, 2003 |publisher=la Repubblica.it |first=Patrizia |last=Capua |language=it}}</ref> an honorary doctorate in economics from the [[University of Antwerp]] in 2007, an honorary doctorate of science from the [[City University of Hong Kong]] in 2011,<ref name = cs-slate-2001-12>{{Cite news |first=Chris |last=Suellentrop |title=A Real Number |url=http://www.slate.com/articles/arts/culturebox/2001/12/a_real_number.single.html |magazine=[[Slate (magazine)|Slate]] |date=December 21, 2001| archive-url = https://web.archive.org/web/20140104104531/http://www.slate.com/articles/arts/culturebox/2001/12/a_real_number.single.html| archive-date = January 4, 2014| url-status = live| access-date = May 28, 2015 |quote=''A Beautiful Mind's'' John Nash is nowhere near as complicated as the real one.}}</ref> and was keynote speaker at a conference on game theory.<ref>{{cite web|title=Nobel Laureate John Nash to Visit HK|url=http://www.china.org.cn/english/scitech/55597.htm|website=china.org.cn|access-date=January 7, 2017}}</ref> Nash also received honorary doctorates from two West Virginia colleges: the University of Charleston in 2003 and West Virginia University Tech in 2006. He was a prolific guest speaker at a number of events, such as the Warwick Economics Summit in 2005, at the [[University of Warwick]]. Nash was elected to the [[American Philosophical Society]] in 2006<ref>{{Cite web|title=APS Member History |url=https://search.amphilsoc.org/memhist/search?creator=John+F.+Nash&title=&subject=&subdiv=&mem=&year=&year-max=&dead=&keyword=&smode=advanced |access-date=May 25, 2021|website=search.amphilsoc.org}}</ref> and became a fellow of the American Mathematical Society in 2012.<ref>[https://www.ams.org/profession/fellows-list List of Fellows of the American Mathematical Society]. Retrieved February 24, 2013.</ref> 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.<ref>{{cite web |title=2015: Nash and Nirenberg |url=https://abelprize.no/abel-prize-laureates/2015 |access-date=August 2, 2022 |website=abelprize.no }}</ref> == Personal life == In 1951, the [[Massachusetts Institute of Technology]] (MIT) hired Nash as a [[C. L. E. Moore instructor]] in the mathematics faculty. About a year later, Nash began a relationship with Eleanor Stier, a nurse he met while admitted as a patient. They had a son, John David Stier,<ref name = cs-slate-2001-12 /> but Nash left Stier when she told him of her pregnancy.<ref>Goldstein, Scott (April 10, 2005) [https://www.boston.com/news/globe/obituaries/articles/2005/04/10/eleanor_stier_84_brookline_nurse_had_son_with_nobel_laureate_mathematician_john_f_nash_jr/ Eleanor Stier, 84; Brookline nurse had son with Nobel laureate mathematician John F. Nash Jr.], Boston.com News.</ref> The film based on Nash's life, ''A Beautiful Mind'', was criticized during the run-up to the 2002 Oscars for omitting this aspect of his life. He was said to have abandoned her based on her social status, which he thought to have been beneath his.<ref>Sutherland, John (March 18, 2002) [https://www.theguardian.com/culture/2002/mar/18/awardsandprizes.highereducation "Beautiful mind, lousy character"], ''The Guardian'', March 18, 2002.</ref> In [[Santa Monica, California]], in 1954, while in his twenties, Nash was arrested for [[indecent exposure]] in a sting operation targeting gay men.<ref>{{ cite news |title=John Nash, mathematician – obituary |url=https://www.telegraph.co.uk/news/obituaries/11627306/John-Nash-mathematician-obituary.html |archive-url=https://ghostarchive.org/archive/20220111/https://www.telegraph.co.uk/news/obituaries/11627306/John-Nash-mathematician-obituary.html |archive-date=January 11, 2022 |url-access=subscription |url-status=live |newspaper=The Telegraph |date=May 24, 2015 |access-date=August 29, 2016}}{{cbignore}}</ref> Although the charges were dropped, he was stripped of his top-secret [[security clearance]] and fired from [[RAND Corporation]], where he had worked as a consultant.<ref>{{cite news |first=Sylvia |last=Nasar |author-link=Sylvia Nasar |title=The sum of a man |url=https://www.theguardian.com/books/2002/mar/26/biography.highereducation |quote=Contrary to widespread references to Nash's "numerous homosexual liaisons", he was not gay. While he had several emotionally intense relationships with other men when he was in his early 20s, I never interviewed anyone who claimed, much less provided evidence, that Nash ever had sex with another man. Nash was arrested in a police trap in a public lavatory in Santa Monica in 1954, at the height of the McCarthy hysteria. The military think-tank where he was a consultant, stripped him of his top-secret security clearance and fired him ... The charge – indecent exposure – was dropped. |newspaper=[[The Guardian]] |date=March 25, 2002 |access-date=July 9, 2012}}</ref> Not long after breaking up with Stier, Nash met [[Alicia Nash|Alicia Lardé Lopez-Harrison]], a [[United States nationality law|naturalized U.S. citizen]] from [[El Salvador]]. Lardé was graduated from [[Massachusetts Institute of Technology|MIT]], having majored in physics.<ref name="Nash1995" /> They married in February 1957. Although Nash was an [[Atheism|atheist]],<ref name="Sylvia Nasar 2011 143">[[#CITEREFNasar1998|Nasar (2011)]], Chapter 17: Bad Boys, p. 143: "In this circle, Nash learned to make a virtue of necessity, styling himself self-consciously as a "free thinker." He announced that he was an atheist."</ref> the ceremony was performed in an [[Episcopal Church (United States)|Episcopal church]].<ref name="charlesmartin">{{Cite web|last=Livio|first=Susan K. |date=June 11, 2017|title=Son of 'A Beautiful Mind' John Nash has one regret |url=https://www.nj.com/healthfit/2017/06/two_years_after_parents_death_son_of_a_beautiful_m.html |access-date=June 17, 2020 |website=NJ Advance Media|language=en}}</ref> In 1958, Nash was appointed to a tenured position at MIT, and his first signs of mental illness soon became evident. He resigned his position at MIT in the spring of 1959.<ref name="Nash1995" /> His son, John Charles Martin Nash, was born a few months later. The child was not named for a year<ref name = cs-slate-2001-12 /> because Alicia felt that Nash should have a say in choosing the name. Due to the stress of dealing with his illness, Nash and Lardé divorced in 1963. After his final hospital discharge in 1970, Nash lived in Lardé's house as a [[Boarding house|boarder]]. This stability seemed to help him, and he learned how to consciously discard his paranoid [[delusion]]s.<ref name='david'>[https://www.nytimes.com/books/98/06/07/daily/mind-book-review.html David Goodstein, 'Mathematics to Madness, and Back'], ''The New York Times'', June 11, 1998</ref> Princeton allowed him to audit classes. He continued to work on mathematics and was eventually allowed to teach again. In the 1990s, Lardé and Nash resumed their relationship, remarrying in 2001. John Charles Martin Nash earned a PhD in mathematics from [[Rutgers University]] and was diagnosed with [[schizophrenia]] as an adult.<ref name="charlesmartin" /> == Death == On May 23, 2015, Nash and his wife died in a car accident on the [[New Jersey Turnpike]] in [[Monroe Township, Middlesex County, New Jersey|Monroe Township, New Jersey]] while returning home from receiving the [[Abel Prize]] in Norway. The driver of the taxicab they were riding in from Newark Airport lost control of the cab and struck a guardrail. Both passengers were ejected and killed.<ref>[https://patch.com/new-jersey/woodbridge/report-cabbie-involved-fatal-turnpike-crash-began-driving-taxi-two-weeks-ago-0 Cabbie In Crash That Killed John Nash Began Driving Taxi Two Weeks Ago]</ref> At the time of his death, Nash was a longtime resident of New Jersey. He was survived by two sons, John Charles Martin Nash, who lived with his parents at the time of their death, and elder child John Stier.<ref>{{cite news |url=http://business.highbeam.com/62734/article-1P1-51401334/john-forbes-nash-may-lose-nj-home |archive-url=https://web.archive.org/web/20130518131511/http://business.highbeam.com/62734/article-1P1-51401334/john-forbes-nash-may-lose-nj-home |archive-date=May 18, 2013 |title=John Forbes Nash May Lose N.J. Home |agency=[[Associated Press]] |date=March 14, 2002 |quote=West Windsor, N.J.: John Forbes Nash Jr., whose life is chronicled in the Oscar-nominated movie ''A Beautiful Mind,'' could lose his home if the township picks one of its proposals to replace a nearby bridge. |access-date=February 22, 2011 |via=[[HighBeam Research]]}}</ref> Following his death, obituaries appeared in scientific and popular media throughout the world. In addition to their obituary for Nash,<ref name="NYT death">{{cite news |first=Erica |last=Goode |url=https://www.nytimes.com/2015/05/25/science/john-nash-a-beautiful-mind-subject-and-nobel-winner-dies-at-86.html |title=John F. Nash Jr., Math Genius Defined by a 'Beautiful Mind,' Dies at 86 |work=[[The New York Times]] |date=May 24, 2015 |access-date=May 24, 2015}}</ref> ''[[The New York Times]]'' published an article containing quotes from Nash that had been assembled from media and other published sources. The quotes consisted of Nash's reflections on his life and achievements.<ref name="NYT quotes">{{cite news |title=The Wisdom of a Beautiful Mind |url=https://www.nytimes.com/interactive/2015/05/25/science/john-nash-quotes.html |website=[[The New York Times]] |access-date=May 25, 2015 |date=May 24, 2015}}</ref> == Legacy == At Princeton in the 1970s, Nash became known as "The Phantom of Fine Hall"<ref>{{cite news |last=Kwon |first=Ha Kyung |title=Nash GS '50: 'The Phantom of Fine Hall' |url=http://dailyprincetonian.com/news/2010/12/nash-gs-50-the-phantom-of-fine-hall/ |access-date=May 6, 2014 |newspaper=[[The Daily Princetonian]] |date=December 10, 2010 |archive-url=https://web.archive.org/web/20140506091547/http://dailyprincetonian.com/news/2010/12/nash-gs-50-the-phantom-of-fine-hall/ |archive-date=May 6, 2014 }}</ref> (Princeton's mathematics center), a shadowy figure who would scribble arcane equations on blackboards in the middle of the night. He is referred to in a novel set at Princeton, ''The Mind-Body Problem'', 1983, by [[Rebecca Goldstein]].<ref name="Nasar1994">{{cite news |last1=Nasar |first1=Sylvia |author-link1=Sylvia Nasar |title=The Lost Years of a Nobel Laureate |url=https://www.nytimes.com/1994/11/13/business/the-lost-years-of-a-nobel-laureate.html |website=[[The New York Times]] |access-date=May 6, 2014 |location=Princeton, New Jersey |date=November 13, 1994}}</ref> [[Sylvia Nasar]]'s biography of Nash, ''[[A Beautiful Mind (book)|A Beautiful Mind]]'', was published in 1998. A [[A Beautiful Mind (film)|film by the same name]] was released in 2001, directed by [[Ron Howard]] with [[Russell Crowe]] playing Nash; it won four [[Academy Awards]], including [[Academy Award for Best Picture|Best Picture]]. For his performance as Nash, Crowe won the [[Golden Globe Award for Best Actor – Motion Picture Drama]] at the [[59th Golden Globe Awards]] and the [[BAFTA Award for Best Actor in a Leading Role|BAFTA Award for Best Actor]] at the [[55th British Academy Film Awards]]. Crowe was nominated for the [[Academy Award for Best Actor]] at the [[74th Academy Awards]]; [[Denzel Washington]] won for his performance in ''[[Training Day]]''. == Awards == * 1978&nbsp;– [[John von Neumann Theory Prize|INFORMS John von Neumann Theory Prize]] (with [[Carlton Lemke]])<ref>{{cite web|url=https://www.informs.org/Recognizing-Excellence/Award-Recipients/John-F.-Nash|title=John F. Nash|access-date=October 10, 2022|website=[[Institute for Operations Research and the Management Sciences]]}}</ref> "for their outstanding contributions to the theory of games" * 1994&nbsp;– [[Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel]] (with [[John Harsanyi]] and [[Reinhard Selten]])<ref>{{cite web|url=https://www.nobelprize.org/prizes/lists/all-prizes-in-economic-sciences/|access-date=October 10, 2022|title=All prizes in economic sciences|website=The Nobel Prize}}</ref> "for their pioneering analysis of equilibria in the theory of non-cooperative games" * 1999&nbsp;– [[Leroy P. Steele Prize for Seminal Contribution to Research]]<ref name="steele" /> for his 1956 paper "The imbedding problem for Riemannian manifolds" * 2002 class of [[Fellow]]s of the [[Institute for Operations Research and the Management Sciences]]<ref>{{citation|url=https://www.informs.org/Recognizing-Excellence/Fellows/Fellows-Alphabetical-List|title=Fellows: Alphabetical List|publisher=[[Institute for Operations Research and the Management Sciences]]|access-date=October 9, 2019|archive-url=https://web.archive.org/web/20190510220119/https://www.informs.org/Recognizing-Excellence/Fellows/Fellows-Alphabetical-List|archive-date=May 10, 2019}}</ref> * 2010&nbsp;– [[Double Helix Medal]]<ref>{{cite web |website=[[Cold Spring Harbor Laboratory]]|title=John F. Nash Jr.: 2010 Honoree |url=http://www.cshl.edu/DHMD/2010-Honoree-John-F-Nash-Jr.html |access-date=July 16, 2014 |archive-url=https://web.archive.org/web/20141017125106/http://www.cshl.edu/DHMD/2010-Honoree-John-F-Nash-Jr.html |archive-date=October 17, 2014 }}</ref> * 2015&nbsp;– [[Abel Prize]] (with [[Louis Nirenberg]])<ref>{{cite web |title=A 'long awaited recognition': Nash receives Abel Prize for revered work in mathematics|url=https://www.princeton.edu/main/news/archive/S42/72/29C63/|website=Princeton University|date=March 26, 2015 |access-date=October 10, 2022|last=Kelly|first=Morgan|department=Office of Communications}}</ref> "for striking and seminal contributions to the theory of nonlinear partial differential equations and its applications to geometric analysis" == Documentaries and interviews == * {{cite episode|last=Wallace|first=Mike (host)|author-link1=Mike Wallace|date=March 17, 2002|network=[[CBS]]|series=60 Minutes|series-link=60 Minutes|season=34|number=26|title=John Nash's Beautiful Mind}} * {{cite episode|title=A Brilliant Madness|url=https://www.pbs.org/wgbh/americanexperience/films/nash|access-date=October 11, 2022|series=American Experience|series-link=American Experience|network=[[Public Broadcasting Service]]|date=April 28, 2002|transcript=Transcript|transcript-url=https://www.pbs.org/wgbh/americanexperience/films/nash/#transcript|last=Samels|first=Mark (director)}} * {{cite interview|interviewer=Marika Griehsel|last=Nash|first=John|date=September 1–4, 2004|title=John F. Nash Jr.|publisher=Nobel Prize Outreach|url=https://www.nobelprize.org/prizes/economic-sciences/1994/nash/interview/}} * {{cite interview|interviewer=[[Riz Khan]]|last=Nash|first=John|date=December 5, 2009|title=One on One|publisher=[[Al Jazeera English]]}} ({{YouTube|UiWBWwCa1E0|Part 1}}, {{YouTube|ufKIgW9XrCE|Part 2}}) * {{cite magazine|title=Interview with Abel Laureate John F. Nash Jr.|interviewer=Martin Raussen and Christian Skau|date=September 2015|magazine=Newsletter of the European Mathematical Society|volume=97|year=2015|pages=26–31|url=https://ems.press/content/serial-issue-files/13732|issn=1027-488X|mr=3409221}} == Publication list == {{refbegin|30em}} * {{cite journal|last1=Nash|last2=Nash|first1=John F.|first2=John F. Jr.|title=Sag and tension calculations for cable and wire spans using catenary formulas|volume=64|issue=10|pages=685–692|doi=10.1109/T-AIEE.1945.5059021|year=1945|journal=[[Transactions of the American Institute of Electrical Engineers]]|s2cid=51640174 }} *{{cite journal|mr=0035977|last1=Nash|first1=John F. Jr.|title=The bargaining problem|journal=[[Econometrica]]|volume=18|year=1950a|pages=155–162|issue=2|doi=10.2307/1907266|jstor=1907266 |zbl=1202.91122|s2cid=153422092}} *{{cite journal|mr=0031701|last1=Nash|first1=John F. Jr.|title=Equilibrium points in {{mvar|n}}-person games|journal=[[Proceedings of the National Academy of Sciences of the United States of America]]|volume=36|year=1950b|pages=48–49|doi=10.1073/pnas.36.1.48|issue=1|pmid=16588946 |pmc=1063129 |bibcode=1950PNAS...36...48N |doi-access=free|zbl=0036.01104}} *{{cite conference|mr=0039223|last1=Nash|first1=J. F.|last2=Shapley|first2=L. S.|author-link2=Lloyd Shapley|title=A simple three-person poker game|zbl=0041.25602|book-title=Contributions to the Theory of Games, Volume I|pages=105–116|series=Annals of Mathematics Studies|volume=24|publisher=[[Princeton University Press]]|location=Princeton, NJ|year=1950|doi=10.1515/9781400881727-011|editor-last1=Kuhn|editor-last2=Tucker|editor-first1=H. W.|editor-first2=A. W.|editor-link2=Albert W. Tucker|editor-link1=Harold W. Kuhn}} *{{cite journal|mr=0043432|last1=Nash|first1=John|title=Non-cooperative games|journal=[[Annals of Mathematics]]|series=Second Series|volume=54|year=1951|pages=286–295|doi=10.2307/1969529|issue=2|jstor=1969529 |zbl=0045.08202}} * {{cite conference|title=Algebraic approximations of manifolds|last1=Nash|first1=John|pages=516–517|url=https://www.mathunion.org/icm/proceedings|book-title=Proceedings of the International Congress of Mathematicians: Cambridge, Massachusetts, U. S. A., 1950. Volume I|year=1952a|location=Providence, RI|publisher=[[American Mathematical Society]]|editor-first1=Lawrence M.|editor-last1=Graves|editor-first3=Paul A.|editor-last3=Smith|editor-first2=Einar|editor-last2=Hille|editor-first4=Oscar|editor-last4=Zariski|editor-link4=Oscar Zariski|editor-link2=Einar Hille|editor-link3=Paul Althaus Smith}} * {{cite journal|mr=0050928|last1=Nash|first1=John|title=Real algebraic manifolds|journal=[[Annals of Mathematics]]|series=Second Series|volume=56|year=1952b|pages=405–421|issue=3|doi=10.2307/1969649|jstor=1969649 |zbl=0048.38501}} * {{cite journal|mr=0053471|last1=Nash|first1=John|title=Two-person cooperative games|journal=[[Econometrica]]|volume=21|year=1953|pages=128–140|doi=10.2307/1906951|issue=1|jstor=1906951 |zbl=0050.14102}} * {{cite journal|mr=3363438|last1=Mayberry|first1=J. P.|last2=Nash|first2=J. F.|last3=Shubik|first3=M.|author-link3=Martin Shubik|title=A comparison of treatments of a duopoly situation|journal=[[Econometrica]]|volume=21|year=1953|issue=1|pages=141–154|doi=10.2307/1906952|jstor=1906952 |zbl=0050.15104|s2cid=154750660 }} * {{cite journal|mr=0065993|last1=Nash|first1=John|title=C<sup>1</sup> isometric imbeddings|journal=[[Annals of Mathematics]]|series=Second Series|volume=60|year=1954|pages=383–396|issue=3|doi=10.2307/1969840|jstor=1969840 |zbl=0058.37703}} * {{cite conference|mr=3363439|last1=Kalisch|first1=G. K.|last2=Milnor|first2=J. W.|author-link2=John Milnor|last3=Nash|first3=J. F.|last4=Nering|first4=E. D.|title=Some experimental {{mvar|n}}-person games|book-title=Decision Processes|pages=301–327|publisher=[[John Wiley & Sons, Inc.]]|location=New York|year=1954|zbl=0058.13904|editor-last1=Thrall|editor-last2=Coombs|editor-last3=Davis|editor-first1=R. M.|editor-first2=C. H.|editor-first3=R. L.|editor-link1=Robert M. Thrall|editor-link2=Clyde Coombs}} * {{cite journal|mr=0071081|last1=Nash|first1=John|title=A path space and the Stiefel–Whitney classes|journal=[[Proceedings of the National Academy of Sciences of the United States of America]]|volume=41|year=1955|pages=320–321|issue=5|doi=10.1073/pnas.41.5.320|pmid=16589673 |pmc=528087 |bibcode=1955PNAS...41..320N |doi-access=free|zbl=0064.17503}} * {{cite journal|mr=0075639|last1=Nash|first1=John|title=The imbedding problem for Riemannian manifolds|journal=[[Annals of Mathematics]]|series=Second Series|volume=63|year=1956|pages=20–63|issue=1|doi=10.2307/1969989|jstor=1969989 |zbl=0070.38603}} * {{cite journal|mr=0089986|last1=Nash|first1=John|title=Parabolic equations|journal=[[Proceedings of the National Academy of Sciences of the United States of America]]|volume=43|year=1957|pages=754–758|issue=8|doi=10.1073/pnas.43.8.754|pmid=16590082 |pmc=528534 |bibcode=1957PNAS...43..754N |doi-access=free|zbl=0078.08704}} * {{cite journal|mr=0100158|last1=Nash|first1=J.|title=Continuity of solutions of parabolic and elliptic equations|journal=[[American Journal of Mathematics]]|volume=80|year=1958|pages=931–954|issue=4|doi=10.2307/2372841|jstor=2372841 |bibcode=1958AmJM...80..931N |zbl=0096.06902}} * {{cite journal|mr=0149094|last1=Nash|first1=John|title=Le problème de Cauchy pour les équations différentielles d'un fluide général|journal=[[Bulletin de la Société Mathématique de France]]|volume=90|year=1962|pages=487–497|zbl=0113.19405|doi=10.24033/bsmf.1586|doi-access=free}} * {{cite journal|mr=0205266|last1=Nash|first1=J.|title=Analyticity of the solutions of implicit function problems with analytic data|journal=[[Annals of Mathematics]]|series=Second Series|volume=84|year=1966|pages=345–355|issue=3|doi=10.2307/1970448|jstor=1970448 |zbl=0173.09202}} * {{cite journal|mr=1381967|last1=Nash|first1=John F. Jr.|title=Arc structure of singularities|journal=[[Duke Mathematical Journal]]|volume=81|year=1995|issue=1|pages=31–38|doi=10.1215/S0012-7094-95-08103-4|zbl=0880.14010}} * {{cite journal|last1=Nash|first1=John|year=2002a|title=Ideal money|journal=[[Southern Economic Journal]]|volume=69|issue=1|pages=4–11|doi=10.2307/1061553|jstor=1061553 }} * {{cite journal|mr=2510706|last1=Nash|first1=John F. Jr.|title=The agencies method for modeling coalitions and cooperation in games|journal=[[International Game Theory Review]]|volume=10|year=2008|issue=4|pages=539–564|doi=10.1142/S0219198908002084|zbl=1178.91019}} * {{cite conference|mr=2605109|last1=Nash|first1=John F.|title=Ideal money and asymptotically ideal money|book-title=Contributions to Game Theory and Management. Volume II|pages=281–293|publisher=Graduate School of Management, [[St. Petersburg University]]|location=St. Petersburg|year=2009a|zbl=1184.91147|isbn=978-5-9924-0020-5|editor-first1=Leon A.|editor-last1=Petrosjan|editor-first2=Nikolay A.|editor-last2=Zenkevich|editor-link1=Leon Petrosyan}} * {{cite journal|mr=2642155|last1=Nash|first1=John F.|title=Studying cooperative games using the method of agencies|journal=International Journal of Mathematics, Game Theory, and Algebra|volume=18|year=2009b|issue=4–5|pages=413–426|zbl=1293.91015}} * {{cite journal|last1=Nash|first1=John F. Jr.|last2=Nagel|first2=Rosemarie|author2-link=Rosemarie Nagel|last3=Ockenfels|first3=Axel|last4=Selten|first4=Reinhard|title=The agencies method for coalition formation in experimental games|journal=[[Proceedings of the National Academy of Sciences of the United States of America]]|volume=109|issue=50|year=2012|pages=20358–20363|doi=10.1073/pnas.1216361109|pmid=23175792 |pmc=3528550 |bibcode=2012PNAS..10920358N |doi-access=free|author-link4=Reinhard Selten|author-link3=Axel Ockenfels}} * {{cite book |editor-last1=Nash | editor-first1=John Forbes Jr. | editor-last2=Rassias | editor-first2=Michael Th. | title=Open problems in mathematics|title-link=Open Problems in Mathematics | publisher=[[Springer Science+Business Media|Springer]]|location=New York | year=2016|mr=3470099|isbn=978-3-319-32160-8|doi=10.1007/978-3-319-32162-2|zbl=1351.00027}} {{refend}} Four of Nash's game-theoretic papers {{harvs|last=Nash|year=1950a|year2=1950b|year3=1951|year4=1953}} and three of his [[pure mathematics]] papers {{harvs|last=Nash|year=1952b|year2=1956|year3=1958}} were collected in the following: * {{cite encyclopedia|mr=1888522|title=The essential John Nash|editor-first1=Harold W.|editor-last1=Kuhn|editor-first2=Sylvia|editor-last2=Nasar|publisher=[[Princeton University Press]]|location=Princeton, NJ|year=2002|isbn=0-691-09527-2|doi=10.1515/9781400884087|editor-link1=Harold Kuhn|editor-link2=Sylvia Nasar|zbl=1033.01024}} == References == {{Reflist}} == Bibliography == * {{cite book|last=Nasar|first=Sylvia|author-link=Sylvia Nasar|title=A Beautiful Mind|url=https://archive.org/details/beautifulmind00sylv|url-access=registration|year=1998|publisher=[[Simon and Schuster]]|isbn=978-1-4391-2649-3|location=New York}} * {{cite book|last=Nasar|first=Sylvia|author-link=Sylvia Nasar|editor-last=Kuhn|editor-first=Harold W.|editor-link=Harold W. Kuhn|title=The Essential John Nash|chapter-url=https://archive.org/details/essentialjohnnas00john|chapter-url-access=registration|year=2002|publisher=[[Princeton University Press]]|isbn=978-0-691-09610-0|ref=SNasar|pages=xi–xxv|jstor=j.ctt1c3gwz0|chapter=Introduction|location=Princeton}} * {{cite book|last=Siegfried|first=Tom|title=A Beautiful Math|year=2006|publisher=[[Joseph Henry Press]]|isbn=978-0-309-10192-9|location=Washington, D.C.}} * {{MacTutor|id=Nash}} == External links == {{Sister project links|d=Q128736|commons=category:John Forbes Nash|m=no|mw=no|species=no|voy=no|s=no|wikt=no|n=no|b=no|v=no}} * [https://web.math.princeton.edu/jfnj/ Home Page of John F. Nash Jr. at Princeton] * {{MathGenealogy|id=18590}} * [https://ideas.repec.org/e/pna13.html IDEAS/RePEc] * [http://www.slate.com/articles/news_and_politics/the_earthling/2002/03/nash_equilibrium.single.html "Nash Equilibrium"] 2002 ''[[Slate (magazine)|Slate]]'' article by [[Robert Wright (journalist)|Robert Wright]], about Nash's work and world government * [http://www.nsa.gov/public_info/press_room/2012/nash_exhibit.shtml NSA releases Nash Encryption Machine plans] {{Webarchive|url=https://web.archive.org/web/20120219190420/http://www.nsa.gov/public_info/press_room/2012/nash_exhibit.shtml |date=February 19, 2012 }} to [[National Cryptologic Museum]] for public viewing, 2012 * {{cite encyclopedia |title=John F. Nash Jr. (1928–2015) |url=http://www.econlib.org/library/Enc/bios/Nash.html |encyclopedia=[[The Concise Encyclopedia of Economics]] |edition=2nd |series=[[Library of Economics and Liberty]] |publisher=[[Liberty Fund]] |year=2016}} * [http://rbsc.princeton.edu/topics/nash-john-1928 Nash, John (1928–2015) | Rare Books and Special Collections] from Princeton's Mudd Library, including a copy of [https://web.archive.org/web/20150524164314/http://www.princeton.edu/mudd/news/faq/topics/Non-Cooperative_Games_Nash.pdf his dissertation] [[Portable Document Format|(PDF)]] * [https://www.informs.org/content/view/full/271160 Biography of John Forbes Nash Jr.] from the Institute for Operations Research and the Management Sciences * {{Nobelprize|namn=John F. Nash Jr.}} {{s-start}} {{s-ach|aw}} {{s-bef | before = [[Robert W. Fogel]] | before2 = [[Douglass C. North]] }} {{s-ttl | title = [[List of Nobel Memorial Prize laureates in Economics|Laureate of the Nobel Memorial Prize in Economics]] | years = 1994 |alongside= [[John C. Harsanyi]], [[Reinhard Selten]]}} {{s-aft | after = [[Robert E. Lucas Jr.]] }} {{s-end}} {{Nobel laureates in economics|state=autocollapse}} {{Abel Prize laureates|state=autocollapse}} {{John von Neumann Theory Prize recipients}} {{Authority control}} {{DEFAULTSORT:Nash, John Forbes Jr.}} [[Category:1928 births]] [[Category:2015 deaths]] [[Category:20th-century American mathematicians]] [[Category:Abel Prize laureates]] [[Category:American atheists]] [[Category:American Nobel laureates]] [[Category:American board game designers]] [[Category:Carnegie Mellon University alumni]] [[Category:Institute for Advanced Study visiting scholars]] [[Category:Differential geometers]] [[Category:Fellows of the American Mathematical Society]] [[Category:Fellows of the Econometric Society]] [[Category:Fellows of the Institute for Operations Research and the Management Sciences]] [[Category:American game theorists]] [[Category:John von Neumann Theory Prize winners]] [[Category:Massachusetts Institute of Technology School of Science faculty]] [[Category:Members of the United States National Academy of Sciences]] [[Category:Nobel laureates in Economics]] [[Category:PDE theorists]] [[Category:People from Bluefield, West Virginia]] [[Category:People from West Windsor, New Jersey]] [[Category:People with schizophrenia]] [[Category:Road incident deaths in New Jersey]] [[Category:Princeton University alumni]] [[Category:Princeton University faculty]] [[Category:Mathematicians from West Virginia]] [[Category:Mathematicians from New Jersey]] [[Category:McLean Hospital patients]] [[Category:Members of the American Philosophical Society]] [[Category:American scientists with disabilities]] </textarea><div class="templatesUsed"><div class="mw-templatesUsedExplanation"><p><span id="templatesused">Pages transcluded onto the current version of this page<span class="posteditwindowhelplinks"> (<a href="/wiki/Help:Transclusion" title="Help:Transclusion">help</a>)</span>:</span> </p></div><ul> <li><a href="/wiki/Template:%27%22" class="mw-redirect" 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