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John Forbes Nash Jr.

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John Forbes Nash Jr.
Nash in the 2000s
Born(1928-06-13)June 13, 1928
Bluefield, West Virginia,U.S.
DiedMay 23, 2015(2015-05-23)(aged 86)
Monroe Township, New Jersey,U.S.
Education
Known for
Spouses
  • ​
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    (m.1957;div.1963)​
  • ​
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    (m.2001; their deaths 2015)​
Awards
Scientific career
Fields
Institutions
ThesisNon-Cooperative Games(1950)
Doctoral advisorAlbert W. Tucker

John Forbes Nash, Jr.(June 13, 1928 – May 23, 2015), known and published asJohn Nash,was an American mathematician who made fundamental contributions togame theory,real algebraic geometry,differential geometry,andpartial differential equations.[1][2]Nash and fellow game theoristsJohn HarsanyiandReinhard Seltenwere awarded the 1994Nobel Memorial Prize in Economics.In 2015, he andLouis Nirenbergwere awarded theAbel Prizefor their contributions to the field of partial differential equations.

As a graduate student in thePrinceton University Department of Mathematics,Nash introduced a number of concepts (includingNash equilibriumand theNash bargaining solution) which are now considered central to game theory and its applications in various sciences. In the 1950s, Nash discovered and proved theNash embedding theoremsby solving a system of nonlinear partial differential equations arising inRiemannian geometry.This work, also introducing a preliminary form of theNash–Moser theorem,was later recognized by theAmerican Mathematical Societywith theLeroy P. Steele Prize for Seminal Contribution to Research.Ennio De Giorgiand Nash found, with separate methods, a body of results paving the way for a systematic understanding ofellipticandparabolic partial differential equations.Their De Giorgi–Nash theorem on the smoothness of solutions of such equations resolvedHilbert's nineteenth problemon regularity in thecalculus of variations,which had been a well-knownopen problemfor almost sixty years.

In 1959, Nash began showing clear signs of mental illness, and spent several years atpsychiatric hospitalsbeing treated forschizophrenia.After 1970, his condition slowly improved, allowing him to return to academic work by the mid-1980s.[3]Nash was biographed inSylvia Nasar's 1998 bookA Beautiful Mind,and his struggles with his illness and his recovery became the basis for afilm of the same namedirected byRon Howard,in which Nash was portrayed byRussell Crowe.[4][5][6]

Early life and education

John Forbes Nash Jr. was born on June 13, 1928, inBluefield, West Virginia.His father and namesake, John Forbes Nash Sr., was anelectrical engineerfor theAppalachian Electric Power Company.His mother, Margaret Virginia (née Martin) Nash, had been a schoolteacher before she was married. He was baptized in theEpiscopal Church.[7]He had a younger sister, Martha (born November 16, 1930).[8]

Nash attended kindergarten and public school, and he learned from books provided by his parents and grandparents.[8]Nash's parents pursued opportunities to supplement their son's education, and arranged for him to take advanced mathematics courses at a local community college (Bluefield College) during his final year of high school. He attendedCarnegie Institute of Technology(which later became Carnegie Mellon University) through a full benefit of the George Westinghouse Scholarship, initially majoring inchemical engineering.He switched to achemistrymajor and eventually, at the advice of his teacherJohn Lighton Synge,to mathematics. After graduating in 1948, with both aB.S.andM.S.in mathematics, Nash accepted a fellowship toPrinceton University,where he pursued furthergraduate studiesin mathematics and sciences.[8]

Nash's adviser and former Carnegie professorRichard Duffinwrote a letter of recommendation for Nash's entrance to Princeton stating, "He is a mathematical genius."[9][10]Nash was also accepted atHarvard University.However, the chairman of the mathematics department at Princeton,Solomon Lefschetz,offered him theJohn S. Kennedyfellowship, convincing Nash that Princeton valued him more.[11]Further, he considered Princeton more favorably because of its proximity to his family in Bluefield.[8]At Princeton, he began work on his equilibrium theory, later known as theNash equilibrium.[12]

Research contributions

Nash in November 2006 at agame theoryconference inCologne,Germany

Nash did not publish extensively, although many of his papers are considered landmarks in their fields.[13]As a graduate student at Princeton, he made foundational contributions togame theoryandreal algebraic geometry.As a postdoctoral fellow atMIT,Nash turned todifferential 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 themathematical analysisofpartial differential equations.[14]After proving his twoisometric embedding theorems,Nash turned to research dealing directly with partial differential equations, where he discovered and proved the De Giorgi–Nash theorem, thereby resolving one form ofHilbert's nineteenth problem.

In 2011, theNational Security Agencydeclassified letters written by Nash in the 1950s, in which he had proposed a newencryption–decryption machine.[15]The letters show that Nash had anticipated many concepts of moderncryptography,which are based oncomputational hardness.[16]

Game theory

Nash earned a PhD in 1950 with a 28-page dissertation onnon-cooperative games.[17][18]The thesis, written under the supervision of doctoral advisorAlbert W. Tucker,contained the definition and properties of theNash equilibrium,a crucial concept in non-cooperative games. A version of his thesis was published a year later in theAnnals of Mathematics.[19]In the early 1950s, Nash carried out research on a number of related concepts in game theory, including the theory ofcooperative games.[20]For his work, Nash was one of the recipients of theNobel Memorial Prize in Economic Sciencesin 1994.

Real algebraic geometry

In 1949, while still a graduate student, Nash found a new result in the mathematical field ofreal algebraic geometry.[21]He announced his theorem in a contributed paper at theInternational Congress of Mathematiciansin 1950, although he had not yet worked out the details of its proof.[22]Nash's theorem was finalized by October 1951, when Nash submitted his work to theAnnals of Mathematics.[23]It had been well-known since the 1930s that everyclosedsmooth manifoldisdiffeomorphicto thezero setof some collection ofsmooth functionsonEuclidean space.In his work, Nash proved that those smooth functions can be taken to bepolynomials.[24]This was widely regarded as a surprising result,[21]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 asNash functionandNash manifold,which have since been widely studied in real algebraic geometry.[24][25]Nash's theorem itself was famously applied byMichael ArtinandBarry Mazurto the study ofdynamical systems,by combining Nash's polynomial approximation together withBézout's theorem.[26][27]

Differential geometry

During his postdoctoral position atMIT,Nash was eager to find high-profile mathematical problems to study.[28]FromWarren Ambrose,adifferential geometer,he learned about the conjecture that anyRiemannian manifoldisisometricto asubmanifoldofEuclidean space.Nash's results proving the conjecture are now known as theNash embedding theorems,the second of whichMikhael Gromovhas called "one of the main achievements of mathematics of the twentieth century".[29]

Nash's first embedding theorem was found in 1953.[28]He found that any Riemannian manifold can be isometrically embedded in a Euclidean space by acontinuously differentiablemapping.[30]Nash's construction allows thecodimensionof 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 Kuipersoon found even smaller codimensions, with the improved result often known as theNash–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 ofdifferential geometry,where high differentiability is significant in much of the usual analysis.[31][32]

However, the logic of Nash's work has been found to be useful in many other contexts inmathematical analysis.Starting with work ofCamillo De Lellisand László Székelyhidi, the ideas of Nash's proof were applied for various constructions of turbulent solutions of theEuler equationsinfluid mechanics.[33][34]In the 1970s,Mikhael Gromovdeveloped Nash's ideas into the general framework ofconvex integration,[32]which has been (among other uses) applied byStefan MüllerandVladimír Šverákto construct counterexamples to generalized forms ofHilbert's nineteenth problemin thecalculus of variations.[35]

Nash found the construction of smoothly differentiable isometric embeddings to be unexpectedly difficult.[28]However, after around a year and a half of intensive work, his efforts succeeded, thereby proving the second Nash embedding theorem.[36]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 animplicit function theoremfor isometric embeddings. The usual formulations of the implicit function theorem are inapplicable, for technical reasons related to theloss of regularityphenomena. Nash's resolution of this issue, given by deforming an isometric embedding by anordinary differential equationalong which extra regularity is continually injected, is regarded as a fundamentally novel technique inmathematical analysis.[37]Nash's paper was awarded theLeroy P. Steele Prize for Seminal Contribution to Researchin 1999, where his "most original idea" in the resolution of theloss of regularityissue was cited as "one of the great achievements in mathematical analysis in this century".[14]According to Gromov:[29]

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 toJürgen Moser's extension of Nash's ideas for application to other problems (notably incelestial mechanics), the resulting implicit function theorem is known as theNash–Moser theorem.It has been extended and generalized by a number of other authors, among them Gromov,Richard Hamilton,Lars Hörmander,Jacob Schwartz,andEduard Zehnder.[32][37]Nash himself analyzed the problem in the context ofanalytic functions.[38]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."[28]According to Gromov:[29]

Nash was solving classical mathematical problems, difficult problems, something that nobody else was able to do, not even to imagine how to do it.... 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.

Partial differential equations

While spending time at theCourant Institutein New York City,Louis Nirenberginformed Nash of a well-known conjecture in the field ofelliptic partial differential equations.[39]In 1938,Charles Morreyhad proved a fundamentalelliptic regularityresult for functions of two independent variables, but analogous results for functions of more than two variables had proved elusive. After extensive discussions with Nirenberg andLars Hörmander,Nash was able to extend Morrey's results, not only to functions of more than two variables, but also to the context ofparabolic partial differential equations.[40]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. TheNash inequalitywas a particular result found in the course of his work (the proof of which Nash attributed toElias Stein), which has been found useful in other contexts.[41][42][43][44]

Soon after, Nash learned fromPaul Garabedian,recently returned from Italy, that the then-unknownEnnio De Giorgihad found nearly identical results for elliptic partial differential equations.[39]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 Moserfound 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 theNash–Moser theorem). De Giorgi and Moser's methods became particularly influential over the next several years, through their developments in the works ofOlga Ladyzhenskaya,James Serrin,andNeil Trudinger,among others.[45][46]Their work, based primarily on the judicious choice oftest functionsin theweak formulationof partial differential equations, is in strong contrast to Nash's work, which is based on analysis of theheat kernel.Nash's approach to the De Giorgi–Nash theory was later revisited byEugene FabesandDaniel Stroock,initiating the re-derivation and extension of the results originally obtained from De Giorgi and Moser's techniques.[41][47]

From the fact that minimizers to many functionals in thecalculus of variationssolve 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, withPeter Laxdescribing it as a "stroke of genius".[39]Nash would later speculate that had it not been for De Giorgi's simultaneous discovery, he would have been a recipient of the prestigiousFields Medalin 1958.[8]Although the medal committee's reasoning is not fully known, and was not purely based on questions of mathematical merit,[48]archival research has shown that Nash placed third in the committee's vote for the medal, after the two mathematicians (Klaus RothandRené Thom) who were awarded the medal that year.[49]

Mental illness

Although Nash'smental illnessfirst began to manifest in the form ofparanoia,his wife later described his behavior as erratic. Nash thought that all men who wore red ties were part of acommunistconspiracy against him. He mailed letters to embassies in Washington, D.C., declaring that they were establishing a government.[3][50]Nash's psychological issues crossed into his professional life when he gave anAmerican Mathematical Societylecture atColumbia Universityin early 1959. Originally intended to present proof of theRiemann hypothesis,the lecture was incomprehensible. Colleagues in the audience immediately realized that something was wrong.[51]

In April 1959, Nash was admitted toMcLean Hospitalfor one month. Based on his paranoid, persecutorydelusions,hallucinations,and increasingasociality,he was diagnosed withschizophrenia.[52][53]In 1961, Nash was admitted to theNew Jersey State Hospital at Trenton.[54]Over the next nine years, he spent intervals of time inpsychiatric hospitals,where he received bothantipsychoticmedicationsandinsulin shock therapy.[53][55]

Although he sometimes took prescribed medication, Nash later wrote that he did so only under pressure. According to Nash, the filmA Beautiful Mindinaccurately implied he was takingatypical antipsychotics.He attributed the depiction to the screenwriter who was worried about the film encouraging people with mental illness to stop taking their medication.[56]

Nash did not take any medication after 1970, nor was he committed to a hospital ever again.[57]Nash recovered gradually.[58]Encouraged by his then former wife, de 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. De Lardé credits hisrecoveryto maintaining "a quiet life" withsocial support.[3]

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'".[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.[59]During his psychotic phase, Nash alsoreferred to himself in the third personas "Johann von Nassau".[60]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".[61]

Nash reported that he started hearing voices in 1964, then later engaged in a process of consciously rejecting them.[62]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.[63]Eventually, he "intellectually rejected" his "delusionally influenced" and "politically oriented" thinking as a waste of effort.[8]In 1995, he said that he did not realize his full potential due to nearly 30 years of mental illness.[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 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 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 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.[8]

Recognition and later career

Nash pictured in 2011

In 1978, Nash was awarded theJohn von Neumann Theory Prizefor his discovery of non-cooperative equilibria, now called Nash Equilibria. He won theLeroy P. Steele Prizein 1999.

In 1994, he received theNobel Memorial Prize in Economic Sciences(along withJohn HarsanyiandReinhard Selten) for hisgame theorywork as a Princeton graduate student.[65]In the late 1980s, Nash had begun to use email to gradually link with working mathematicians who realized that he wastheJohn Nash and that his new work had value. They formed part of the nucleus of a group that contacted theBank of Sweden's Nobel award committee and were able to vouch for Nash's mental health and ability to receive the award.[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. 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 "onstrike"from an economic point of view. He advanced views inevolutionary psychologyabout the potential benefits of apparently nonstandard behaviors or roles.[67]

Nash criticizedKeynesian ideasofmonetary economicswhich allowed for acentral bankto implementmonetary policies.[68]He proposed a standard of "Ideal Money" pegged to an "industrial consumptionprice index"which was more stable than" bad money. "He noted that his thinking on money and the function ofmonetary authorityparalleled that of economistFriedrich Hayek.[69][68]

Nash received an honorary degree, Doctor of Science and Technology, fromCarnegie Mellon Universityin 1999, an honorary degree in economics from theUniversity of Naples Federico IIin 2003,[70]an honorary doctorate in economics from theUniversity of Antwerpin 2007, an honorary doctorate of science from theCity University of Hong Kongin 2011,[71]and was keynote speaker at a conference on game theory.[72]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 theUniversity of Warwick.

Nash was elected to theAmerican Philosophical Societyin 2006[73]and became a fellow of the American Mathematical Society in 2012.[74]

On May 19, 2015, a few days before his death, Nash, along withLouis Nirenberg,was awarded the 2015Abel Prizeby KingHarald V of Norwayat a ceremony in Oslo.[75]

Personal life

In 1951, theMassachusetts Institute of Technology(MIT) hired Nash as aC. L. E. Moore instructorin 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,[71]but Nash left Stier when she told him of her pregnancy.[76]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.[77]

InSanta Monica, California,in 1954, while in his twenties, Nash was arrested forindecent exposurein a sting operation targeting gay men.[78]Although the charges were dropped, he was stripped of his top-secretsecurity clearanceand fired fromRAND Corporation,where he had worked as a consultant.[79]

Not long after breaking up with Stier, Nash metAlicia Lardé Lopez-Harrison,anaturalized U.S. citizenfromEl Salvador.Lardé graduated fromMIT,having majored in physics.[8]They married in February 1957. Although Nash was anatheist,[80]the ceremony was performed in anEpiscopal church.[81]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.[8]His son, John Charles Martin Nash, was born a few months later. The child was not named for a year[71]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 aboarder.This stability seemed to help him, and he learned how to consciously discard his paranoiddelusions.[82]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 fromRutgers Universityand was diagnosed withschizophreniaas an adult.[81]

Death

On May 23, 2015, Nash and his wife died in a car accident on theNew Jersey TurnpikeinMonroe Township, New Jerseywhile returning home from receiving theAbel Prizein 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.[83]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.[84]

Following his death, obituaries appeared in scientific and popular media throughout the world. In addition to their obituary for Nash,[85]The New York Timespublished 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.[86]

Legacy

At Princeton in the 1970s, Nash became known as "The Phantom of Fine Hall"[87](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, byRebecca Goldstein.[3]

Sylvia Nasar's biography of Nash,A Beautiful Mind,was published in 1998. Afilm by the same namewas released in 2001, directed byRon HowardwithRussell Croweplaying Nash; it won fourAcademy Awards,includingBest Picture.For his performance as Nash, Crowe won theGolden Globe Award for Best Actor – Motion Picture Dramaat the59th Golden Globe Awardsand theBAFTA Award for Best Actorat the55th British Academy Film Awards.Crowe was nominated for theAcademy Award for Best Actorat the74th Academy Awards;Denzel Washingtonwon for his performance inTraining Day.

Awards

Documentaries and interviews

  • Wallace, Mike (host)(March 17, 2002). "John Nash's Beautiful Mind".60 Minutes.Season 34. Episode 26.CBS.
  • Samels, Mark (director) (April 28, 2002)."A Brilliant Madness".American Experience.Public Broadcasting Service.Transcript.RetrievedOctober 11,2022.
  • Nash, John (September 1–4, 2004)."John F. Nash Jr"(Interview). Interviewed by Marika Griehsel. Nobel Prize Outreach.
  • Nash, John (December 5, 2009). "One on One" (Interview). Interviewed byRiz Khan.Al Jazeera English.(Part 1onYouTube,Part 2onYouTube)
  • "Interview with Abel Laureate John F. Nash Jr".Newsletter of the European Mathematical Society.Vol. 97. Interviewed by Martin Raussen and Christian Skau. September 2015. pp. 26–31.ISSN1027-488X.MR3409221.{{cite magazine}}:CS1 maint: date and year (link)


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Publication list

Four of Nash's game-theoretic papers (Nash1950a,1950b,1951,1953) and three of hispure mathematicspapers (Nash1952b,1956,1958) were collected in the following:

References

  1. ^Goode, Erica (May 24, 2015)."John F. Nash Jr., Math Genius Defined by a 'Beautiful Mind,' Dies at 86".The New York Times.
  2. ^"John F. Nash Jr. and Louis Nirenberg share the Abel Prize".Abel Prize.March 25, 2015. Archived fromthe originalon June 16, 2019.RetrievedMay 27,2015.
  3. ^abcdNasar, Sylvia(November 13, 1994)."The Lost Years of a Nobel Laureate".The New York Times.Princeton, New Jersey.RetrievedMay 6,2014.
  4. ^"Oscar race scrutinizes movies based on true stories".USA Today.March 6, 2002.RetrievedJanuary 22,2008.
  5. ^"Academy Award Winners".USA Today.March 25, 2002.RetrievedAugust 30,2008.
  6. ^Yuhas, Daisy (March 2013)."Throughout History, Defining Schizophrenia Has Remained A Challenge (Timeline)".Scientific American Mind.RetrievedMarch 2,2013.
  7. ^Nasar 1998,Chapter 1.
  8. ^abcdefghijNash, John F. Jr. (1995)."John F. Nash Jr. – Biographical".InFrängsmyr, Tore(ed.).The Nobel Prizes 1994: Presentations, Biographies & Lectures.Stockholm:Nobel Foundation.pp. 275–279.ISBN978-91-85848-24-9.
  9. ^"Nash recommendation letter"(PDF).p. 23. Archived fromthe original(PDF)on June 7, 2017.RetrievedJune 5,2015.
  10. ^Kuhn, Harold W.;Nasar, Sylvia (eds.)."The Essential John Nash"(PDF).Princeton University Press.pp. Introduction, xi.Archived(PDF)from the original on January 1, 2007.RetrievedApril 17,2008.
  11. ^Nasar 1998,Chapter 2.
  12. ^Nasar (2002), pp. xvi–xix.
  13. ^Milnor, John(1998)."John Nash and 'A Beautiful Mind'"(PDF).Notices of the American Mathematical Society.25(10): 1329–1332.
  14. ^abc"1999 Steele Prizes"(PDF).Notices of the American Mathematical Society.46(4): 457–462. April 1999.Archived(PDF)from the original on August 29, 2000.
  15. ^"2012 Press Release – National Cryptologic Museum Opens New Exhibit on Dr. John Nash".National Security Agency.RetrievedJuly 30,2022.
  16. ^"John Nash's Letter to the NSA; Turing's Invisible Hand".February 17, 2012.RetrievedFebruary 25,2012.
  17. ^Nash, John F.(May 1950)."Non-Cooperative Games"(PDF).PhD thesis.Princeton University. Archived fromthe original(PDF)on April 20, 2015.RetrievedMay 24,2015.
  18. ^Osborne, Martin J. (2004).An Introduction to Game Theory.Oxford, England:Oxford University Press.p.23.ISBN0-19-512895-8.
  19. ^Nash 1951.
  20. ^Nash 1950a;Nash 1950b;Nash 1953.
  21. ^abNasar 1998,Chapter 15.
  22. ^Nash 1952a.
  23. ^Nash 1952b.
  24. ^abBochnak, Jacek; Coste, Michel;Roy, Marie-Françoise(1998).Real algebraic geometry.Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge.Vol. 36 (Translated and revised from 1987 French original ed.). Berlin:Springer-Verlag.doi:10.1007/978-3-662-03718-8.ISBN3-540-64663-9.MR1659509.S2CID118839789.Zbl0912.14023.
  25. ^Shiota, Masahiro (1987).Nash Manifolds.Lecture Notes in Mathematics.Vol. 1269. Berlin:Springer-Verlag.doi:10.1007/BFb0078571.ISBN3-540-18102-4.MR0904479.Zbl0629.58002.
  26. ^Artin, M.;Mazur, B.(1965). "On periodic points".Annals of Mathematics.Second Series.81(1): 82–99.doi:10.2307/1970384.JSTOR1970384.MR0176482.Zbl0127.13401.
  27. ^Gromov, Mikhaïl(2003)."On the entropy of holomorphic maps"(PDF).L'Enseignement Mathématique. Revue Internationale.2e Série.49(3–4): 217–235.MR2026895.Zbl1080.37051.
  28. ^abcdNasar 1998,Chapter 20.
  29. ^abcGromov, Misha(2016). "Introduction John Nash: theorems and ideas". In Nash, John Forbes Jr.; Rassias, Michael Th. (eds.).Open problems in mathematics.Springer, Cham.arXiv:1506.05408.doi:10.1007/978-3-319-32162-2.ISBN978-3-319-32160-8.MR3470099.
  30. ^Nash 1954.
  31. ^Eliashberg, Y.;Mishachev, N. (2002).Introduction to the h-principle.Graduate Studies in Mathematics.Vol. 48. Providence, RI:American Mathematical Society.doi:10.1090/gsm/048.ISBN0-8218-3227-1.MR1909245.
  32. ^abcGromov, Mikhael(1986).Partial differential relations.Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Vol. 9. Berlin:Springer-Verlag.doi:10.1007/978-3-662-02267-2.ISBN3-540-12177-3.MR0864505.
  33. ^De Lellis, Camillo;Székelyhidi, László Jr. (2013). "Dissipative continuous Euler flows".Inventiones Mathematicae.193(2): 377–407.arXiv:1202.1751.Bibcode:2013InMat.193..377D.doi:10.1007/s00222-012-0429-9.MR3090182.S2CID2693636.
  34. ^Isett, Philip (2018)."A proof of Onsager's conjecture".Annals of Mathematics.Second Series.188(3): 871–963.arXiv:1608.08301.doi:10.4007/annals.2018.188.3.4.MR3866888.S2CID119267892.Archived fromthe originalon October 11, 2022.RetrievedOctober 11,2022.
  35. ^Müller, S.;Šverák, V.(2003)."Convex integration for Lipschitz mappings and counterexamples to regularity".Annals of Mathematics.Second Series.157(3): 715–742.arXiv:math/0402287.doi:10.4007/annals.2003.157.715.MR1983780.S2CID55855605.
  36. ^Nash 1956.
  37. ^abHamilton, Richard S.(1982)."The inverse function theorem of Nash and Moser".Bulletin of the American Mathematical Society.New Series.7(1): 65–222.doi:10.1090/s0273-0979-1982-15004-2.MR0656198.Zbl0499.58003.
  38. ^Nash 1966.
  39. ^abcNasar 1998,Chapter 30.
  40. ^Nash 1957;Nash 1958.
  41. ^abDavies, E. B.(1989).Heat kernels and spectral theory.Cambridge Tracts in Mathematics. Vol. 92. Cambridge:Cambridge University Press.doi:10.1017/CBO9780511566158.ISBN0-521-36136-2.MR0990239.
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  44. ^Lieb, Elliott H.;Loss, Michael(2001).Analysis.Graduate Studies in Mathematics.Vol. 14 (Second edition of 1997 original ed.). Providence, RI:American Mathematical Society.ISBN0-8218-2783-9.MR1817225.
  45. ^Gilbarg, David;Trudinger, Neil S.(2001).Elliptic partial differential equations of second order.Classics in Mathematics (Reprint of the second ed.). Berlin:Springer-Verlag.doi:10.1007/978-3-642-61798-0.ISBN3-540-41160-7.MR1814364.
  46. ^Lieberman, Gary M. (1996).Second order parabolic differential equations.River Edge, NJ:World Scientific Publishing Co., Inc.doi:10.1142/3302.ISBN981-02-2883-X.MR1465184.
  47. ^Fabes, E. B.;Stroock, D. W.(1986). "A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash".Archive for Rational Mechanics and Analysis.96(4): 327–338.Bibcode:1986ArRMA..96..327F.doi:10.1007/BF00251802.MR0855753.S2CID189774501.
  48. ^Nasar 1998,Chapter 31.
  49. ^Barany, Michael (January 18, 2018)."The Fields Medal should return to its roots".Nature.553(7688): 271–273.Bibcode:2018Natur.553..271B.doi:10.1038/d41586-018-00513-8.
  50. ^Nasar (2011),p. 251.
  51. ^Sabbagh, Karl (2003).Dr. Riemann's Zeros.London, England:Atlantic Books.pp.87–88.ISBN1-84354-100-9.
  52. ^"Brown University Didactic Readings: DSM-IV Schizophrenia (DSM-IV-TR #295.1–295.3, 295.90)"(PDF).Providence, Rhode Island:Brown University.pp. 1–11.RetrievedJune 1,2015.
  53. ^abNasar (2011),p. 32.
  54. ^O'Connor, John J.;Robertson, Edmund F.,"John Forbes Nash Jr.",MacTutor History of Mathematics Archive,University of St Andrews
  55. ^Ebert, Roger(2002).Roger Ebert's Movie Yearbook 2003.Andrews McMeel Publishing.ISBN978-0-7407-2691-0.RetrievedJuly 10,2008.
  56. ^Greihsel, Marika (September 1, 2004)."John F. Nash Jr. – Interview".Nobel Foundation.RetrievedNovember 3,2018.
  57. ^Nash, John Forbes (2002)."PBS Interview: Medication".PBS.Archived fromthe originalon June 4, 2016.RetrievedSeptember 1,2017.
  58. ^Nash, John"PBS Interview: How does Recovery Happen?"ArchivedJune 6, 2016, at theWayback Machine2002.
  59. ^Nash, John"PBS Interview: Delusional Thinking"ArchivedOctober 1, 2016, at theWayback Machine.2002.
  60. ^Nasar 1998,Chapter 39.
  61. ^Nash, John"PBS Interview: The Downward Spiral"ArchivedMarch 10, 2017, at theWayback Machine2002.
  62. ^Nash, John"PBS Interview: Hearing voices"ArchivedMarch 9, 2012, at theWayback Machine.2002.
  63. ^Nash, John"PBS Interview: Paths to Recovery"ArchivedJune 5, 2016, at theWayback Machine.2002.
  64. ^Nash, John"John Nash: My experience with mental illness"ArchivedDecember 7, 2016, at theWayback Machine.PBS Interview, 2002.
  65. ^Nasar (2002), p. xiii.
  66. ^"The Work of John Nash in Game Theory"(PDF).Nobel Seminar.December 8, 1994. Archived fromthe original(PDF)on August 10, 2013.RetrievedMay 29,2015.
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  68. ^abZuckerman, Julia (April 27, 2005)"Nobel winner Nash critiques economic theory".The Brown Daily Herald.By JULIA ZUCKERMAN Wednesday, April 27, 2005
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  71. ^abcSuellentrop, Chris (December 21, 2001)."A Real Number".Slate.Archivedfrom the original on January 4, 2014.RetrievedMay 28,2015.A Beautiful Mind'sJohn Nash is nowhere near as complicated as the real one.
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  74. ^List of Fellows of the American Mathematical Society.Retrieved February 24, 2013.
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  76. ^Goldstein, Scott (April 10, 2005)Eleanor Stier, 84; Brookline nurse had son with Nobel laureate mathematician John F. Nash Jr.,Boston News.
  77. ^Sutherland, John (March 18, 2002)"Beautiful mind, lousy character",The Guardian,March 18, 2002.
  78. ^"John Nash, mathematician – obituary".The Telegraph.May 24, 2015.Archivedfrom the original on January 11, 2022.RetrievedAugust 29,2016.
  79. ^Nasar, Sylvia(March 25, 2002)."The sum of a man".The Guardian.RetrievedJuly 9,2012.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.
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  81. ^abLivio, Susan K. (June 11, 2017)."Son of 'A Beautiful Mind' John Nash has one regret".NJ Advance Media.RetrievedJune 17,2020.
  82. ^David Goodstein, 'Mathematics to Madness, and Back',The New York Times,June 11, 1998
  83. ^Cabbie In Crash That Killed John Nash Began Driving Taxi Two Weeks Ago
  84. ^"John Forbes Nash May Lose N.J. Home".Associated Press.March 14, 2002. Archived fromthe originalon May 18, 2013.RetrievedFebruary 22,2011– viaHighBeam Research.West Windsor, N.J.: John Forbes Nash Jr., whose life is chronicled in the Oscar-nominated movieA Beautiful Mind,could lose his home if the township picks one of its proposals to replace a nearby bridge.
  85. ^Goode, Erica (May 24, 2015)."John F. Nash Jr., Math Genius Defined by a 'Beautiful Mind,' Dies at 86".The New York Times.RetrievedMay 24,2015.
  86. ^"The Wisdom of a Beautiful Mind".The New York Times.May 24, 2015.RetrievedMay 25,2015.
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  89. ^"All prizes in economic sciences".The Nobel Prize.RetrievedOctober 10,2022.
  90. ^Fellows: Alphabetical List,Institute for Operations Research and the Management Sciences,archived fromthe originalon May 10, 2019,retrievedOctober 9,2019
  91. ^"John F. Nash Jr.: 2010 Honoree".Cold Spring Harbor Laboratory.Archived fromthe originalon October 17, 2014.RetrievedJuly 16,2014.
  92. ^Kelly, Morgan (March 26, 2015)."A 'long awaited recognition': Nash receives Abel Prize for revered work in mathematics".Office of Communications.Princeton University.RetrievedOctober 10,2022.

Bibliography

Awards
Preceded by Laureate of the Nobel Memorial Prize in Economics
1994
Served alongside:John C. Harsanyi,Reinhard Selten 		Succeeded by
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