John nash economics contribution

American mathematician (1928–2015)

John Forbes Nash, Jr. (June 13, 1928 – May 23, 2015), customary and published as John Nash, was an American mathematician who made fundamental contributions to pastime theory, real algebraic geometry, discrimination geometry, and partial differential equations.^[1][2] Nash and fellow game theorists John Harsanyi and Reinhard Selten were awarded the 1994 Altruist Prize in Economics. In 2015, he and Louis Nirenberg were awarded the Abel Prize carry their contributions to the specialization of partial differential equations.

As a graduate student in interpretation Princeton University Department of Math, Nash introduced a number describe concepts (including Nash equilibrium unacceptable the Nash bargaining solution) which are now considered central show to advantage game theory and its applications in various sciences. In nobleness 1950s, Nash discovered and compliant the Nash embedding theorems wedge solving a system of nonlinear partial differential equations arising pile Riemannian geometry. This work, as well introducing a preliminary form invoke the Nash–Moser theorem, was ulterior recognized by the American Exact Society with the Leroy Proprietress. Steele Prize for Seminal Impost to Research. Ennio De Giorgi and Nash found, with come between methods, a body of careful paving the way for expert systematic understanding of elliptic squeeze parabolic partial differential equations. Their De Giorgi–Nash theorem on decency smoothness of solutions of much equations resolved Hilbert's nineteenth question on regularity in the rock of variations, which had archaic a well-known open problem pursue almost sixty years.

In 1959, Nash began showing clear symbols of mental illness, and all in several years at psychiatric hospitals being treated for schizophrenia. Subsequently 1970, his condition slowly reinforced, allowing him to return come close to academic work by the mid-1980s.^[3]

Nash's life was the subject avail yourself of Sylvia Nasar's 1998 biographical work A Beautiful Mind, and surmount struggles with his illness tell off his recovery became the rationale for a film of righteousness same name directed by Daffo Howard, in which Nash was portrayed by Russell Crowe.^[4][5][6]

Early strength of mind and education

John Forbes Nash Jr. was born on June 13, 1928, in Bluefield, West Town. His father and namesake, Bathroom Forbes Nash Sr., was pull out all the stops electrical engineer for the Appalachian Electric Power Company. His inactivity, Margaret Virginia (née Martin) Writer, had been a schoolteacher heretofore she was married. He was baptized in the Episcopal Cathedral. He had a younger develop, Martha (born November 16, 1930).^[8]

Nash attended kindergarten and public institution, and he learned from books provided by his parents take precedence grandparents.^[8] Nash's parents pursued opportunities to supplement their son's instruction, and arranged for him work take advanced mathematics courses draw on nearby Bluefield College (now Bluefield University) during his final assemblage of high school. He packed with Carnegie Institute of Technology (which later became Carnegie Mellon University) through a full benefit a few the George Westinghouse Scholarship, at first majoring in chemical engineering. Crystalclear switched to a chemistry older and eventually, at the alert of his teacher John Lighton Synge, to mathematics. After graduating in 1948, with both orderly B.S. and M.S. in maths, Nash accepted a fellowship allude to Princeton University, where he track further graduate studies in math and sciences.^[8]

Nash's adviser and prior Carnegie professor Richard Duffin wrote a letter of recommendation in favour of Nash's entrance to Princeton stating, "He is a mathematical genius."^[9][10] Nash was also accepted try to be like Harvard University. However, the head of the mathematics department defer Princeton, Solomon Lefschetz, offered him the John S. Kennedy companionship, convincing Nash that Princeton esteemed him more. Further, he believed Princeton more favorably because model its proximity to his affinity in Bluefield.^[8] At Princeton, proscribed began work on his balance theory, later known as significance Nash equilibrium.^[12]

Research contributions

Nash did crowd publish extensively, although many be in possession of his papers are considered landmarks in their fields.^[13] As undiluted graduate student at Princeton, unquestionable made foundational contributions to project theory and real algebraic geometry. As a postdoctoral fellow chimpanzee MIT, Nash turned to differentiation geometry. Although the results ad infinitum Nash's work on differential geometry are phrased in a geometric language, the work is partly entirely to do with excellence mathematical analysis of partial perception equations.^[14] After proving his four isometric embedding theorems, Nash profane to research dealing directly market partial differential equations, where perform discovered and proved the Subjective Giorgi–Nash theorem, thereby resolving undeniable form of Hilbert's nineteenth attention.

In 2011, the National Reassurance Agency declassified letters written tough Nash in the 1950s, person of little consequence which he had proposed cool new encryption–decryption machine.^[15] The longhand show that Nash had due many concepts of modern writing, which are based on computational hardness.^[16]

Game theory

Nash earned a PhD in 1950 with a 28-page dissertation on non-cooperative games.^[17][18] Nobility thesis, written under the surveillance of doctoral advisor Albert Unprotected. Tucker, contained the definition opinion properties of the Nash balance, a crucial concept in non-cooperative games. A version of coronet thesis was published a origin later in the Annals apply Mathematics. In the early Decennary, Nash carried out research rule a number of related concepts in game theory, including say publicly theory of cooperative games. Set out his work, Nash was see to of the recipients of primacy Nobel Memorial Prize in Reduced Sciences in 1994.

Real algebraical geometry

In 1949, while still pure graduate student, Nash found first-class new result in the exact field of real algebraic geometry. He announced his theorem meet a contributed paper at authority International Congress of Mathematicians foundation 1950, although he had crowd together yet worked out the trivia of its proof. Nash's postulate was finalized by October 1951, when Nash submitted his snitch to the Annals of Science. It had been well-known because the 1930s that every closedsmooth manifold is diffeomorphic to high-mindedness zero set of some grade of smooth functions on Geometer space. In his work, Writer proved that those smooth functions can be taken to breed polynomials.^[24] This was widely thought as a surprising result, in that the class of smooth functions and smooth manifolds is customarily far more flexible than leadership class of polynomials. Nash's be consistent with introduced the concepts now make public as Nash function and Writer manifold, which have since archaic widely studied in real algebraical geometry.^[24][25] Nash's theorem itself was famously applied by Michael Artin and Barry Mazur to magnanimity study of dynamical systems, offspring combining Nash's polynomial approximation contrive with Bézout's theorem.^[26][27]

Differential geometry

During climax postdoctoral position at MIT, Author was eager to find high-profile mathematical problems to study. Do too much Warren Ambrose, a differential geometrician, he learned about the supposition that any Riemannian manifold recapitulate isometric to a submanifold give a miss Euclidean space. Nash's results proving the conjecture are now be revealed as the Nash embedding theorems, the second of which Mikhael Gromov has called "one more than a few the main achievements of arithmetic of the twentieth century".^[29]

Nash's cardinal embedding theorem was found weighty 1953. He found that considerable Riemannian manifold can be isometrically embedded in a Euclidean timespan by a continuously differentiable planning. Nash's construction allows the codimension of the embedding to subsist very small, with the weekend case that in many cases focus is logically impossible that graceful highly-differentiable isometric embedding exists. (Based on Nash's techniques, Nicolaas Kuiper soon found even smaller codimensions, with the improved result usually known as the Nash–Kuiper theorem.) As such, Nash's embeddings tip limited to the setting pounce on low differentiability. For this basis, Nash's result is somewhat exterior the mainstream in the marker of differential geometry, where soaring differentiability is significant in unnecessary of the usual analysis.^[31][32]

However, picture logic of Nash's work has been found to be worthy in many other contexts rejoinder mathematical analysis. Starting with industry of Camillo De Lellis streak László Székelyhidi, the ideas neat as a new pin Nash's proof were applied make it to various constructions of turbulent solutions of the Euler equations bay fluid mechanics.^[33][34] In the Decennium, Mikhael Gromov developed Nash's matter into the general framework precision convex integration,^[32] which has antique (among other uses) applied soak Stefan Müller and Vladimír Šverák to construct counterexamples to imprecise forms of Hilbert's nineteenth complication in the calculus of variations.^[35]

Nash found the construction of happily differentiable isometric embeddings to remedy unexpectedly difficult. However, after ensemble a year and a portion of intensive work, his efforts succeeded, thereby proving the next Nash embedding theorem. The burden involved in proving this next theorem are largely separate use those used in proving nobility first. The fundamental aspect remaining the proof is an implied function theorem for isometric embeddings. The usual formulations of rectitude implicit function theorem are unlike beside the point, for technical reasons related assent to the loss of regularity phenomena. Nash's resolution of this negligible, given by deforming an line embedding by an ordinary figuring equation along which extra symmetry is continually injected, is purported as a fundamentally novel appeal in mathematical analysis.^[37] Nash's procedure was awarded the Leroy Proprietor. Steele Prize for Seminal Imposition to Research in 1999, veer his "most original idea" rise the resolution of the loss of regularity issue was empty as "one of the fair achievements in mathematical analysis pretend this century".^[14] According to Gromov:^[29]

You must be a novice thorough analysis or a genius with regards to Nash to believe anything need that can be ever speculation and/or to have a nonpareil nontrivial application.

Due to Jürgen Moser's extension of Nash's ideas carry application to other problems (notably in celestial mechanics), the secondary implicit function theorem is overwhelm as the Nash–Moser theorem. Outdo has been extended and blurred by a number of goad authors, among them Gromov, Richard Hamilton, Lars Hörmander, Jacob Schwartz, and Eduard Zehnder.^[32][37] Nash analyzed the problem in excellence context of analytic functions. Schwartz later commented that Nash's text were "not just novel, however very mysterious," and that collection was very hard to "get to the bottom of it." According to Gromov:^[29]

Nash was answer classical mathematical problems, difficult on, something that nobody else was able to do, not yet to imagine how to release it. ...  what Nash unconcealed in the course of realm constructions of isometric embeddings deference far from 'classical' – it crack something that brings about on the rocks dramatic alteration of our encounter of the basic logic look up to analysis and differential geometry. Judgment from the classical perspective, what Nash has achieved in fulfil papers is as impossible importation the story of his life ... [H]is work on isometric immersions ... opened a new world firm footing mathematics that stretches in expansion of our eyes in up till unknown directions and still waits to be explored.

Partial differential equations

While spending time at the Courant Institute in New York Discard, Louis Nirenberg informed Nash rivalry a well-known conjecture in say publicly field of elliptic partial discernment equations. In 1938, Charles Morrey had proved a fundamental elliptical regularity result for functions perceive two independent variables, but correspondent results for functions of further than two variables had worthy elusive. After extensive discussions keep an eye on Nirenberg and Lars Hörmander, Author was able to extend Morrey's results, not only to functions of more than two variables, but also to the situation of parabolic partial differential equations. In his work, as advise Morrey's, uniform control over righteousness continuity of the solutions prospect such equations is achieved, externally assuming any level of differentiability on the coefficients of probity equation. The Nash inequality was a particular result found amuse the course of his effort (the proof of which Writer attributed to Elias Stein), which has been found useful tidy other contexts.^{[41][42][43][44]}

Soon after, Nash cultured from Paul Garabedian, recently reciprocal from Italy, that the then-unknown Ennio De Giorgi had difficult nearly identical results for elliptical partial differential equations. De Giorgi and Nash's methods had tiny to do with one selection, although Nash's were somewhat build on powerful in applying to both elliptic and parabolic equations. Expert few years later, inspired tough De Giorgi's method, Jürgen Moser found a different approach forbear the same results, and significance resulting body of work decay now known as the Observe Giorgi–Nash theorem or the Hilarity Giorgi–Nash–Moser theory (which is definite from the Nash–Moser theorem). Refrain from Giorgi and Moser's methods became particularly influential over the future several years, through their developments in the works of Olga Ladyzhenskaya, James Serrin, and Neil Trudinger, among others.^[45][46] Their exert yourself, based primarily on the reckless choice of test functions squeeze the weak formulation of non-discriminatory differential equations, is in difficult contrast to Nash's work, which is based on analysis holiday the heat kernel. Nash's providing to the De Giorgi–Nash conception was later revisited by General Fabes and Daniel Stroock, prep after the re-derivation and extension go along with the results originally obtained shun De Giorgi and Moser's techniques.^[41][47]

From the fact that minimizers accomplish many functionals in the incrustation of variations solve elliptic passable differential equations, Hilbert's nineteenth dilemma (on the smoothness of these minimizers), conjectured almost sixty adulthood prior, was directly amenable adopt the De Giorgi–Nash theory. Writer received instant recognition for crown work, with Peter Lax reading it as a "stroke innumerable genius". Nash would later jeopardize that had it not back number for De Giorgi's simultaneous bargain, he would have been swell recipient of the prestigious Comedian Medal in 1958.^[8] Although high-mindedness medal committee's reasoning is plead for fully known, and was plead for purely based on questions indicate mathematical merit, archival research has shown that Nash placed position in the committee's vote compel the medal, after the bend in half mathematicians (Klaus Roth and René Thom) who were awarded integrity medal that year.^[49]

Mental illness

Although Nash's mental illness first began hinder manifest in the form catch paranoia, his wife later dubious his behavior as erratic. Writer thought that all men who wore red ties were possessions of a communist conspiracy admit him. He mailed letters equal embassies in Washington, D.C., notice that they were establishing first-class government.^[3][50] Nash's psychological issues intersectant into his professional life during the time that he gave an American Controlled Society lecture at Columbia Establishment in early 1959. Originally lucky break to present proof of loftiness Riemann hypothesis, the lecture was incomprehensible. Colleagues in the introduction immediately realized that something was wrong.^[51]

In April 1959, Nash was admitted to McLean Hospital uncontaminated one month. Based on jurisdiction paranoid, persecutory delusions, hallucinations, professor increasing asociality, he was diagnosed with schizophrenia.^[52][53] In 1961, Author was admitted to the Modern Jersey State Hospital at Trenton.^[54] Over the next nine duration, he spent intervals of date in psychiatric hospitals, where of course received both antipsychoticmedications and insulin shock therapy.^[53][55]

Although he sometimes took prescribed medication, Nash later wrote that he did so sole under pressure. According to Writer, the film A Beautiful Mind inaccurately implied he was engaging atypical antipsychotics. He attributed justness depiction to the screenwriter who was worried about the vinyl encouraging people with mental disease to stop taking their medication.^[56]

Nash did not take any healing after 1970, nor was forbidden committed to a hospital day out again.^[57] Nash recovered gradually.^[58] Pleased by his then former old lady, Lardé, Nash lived at house and spent his time inconvenience the Princeton mathematics department whirl location his eccentricities were accepted uniform when his mental condition was poor. Lardé credits his restoration to maintaining "a quiet life" with social support.^[3]

Nash dated rendering start of what he termed "mental disturbances" to the inconvenient months of 1959, when sovereign wife was pregnant. He stated doubtful a process of change "from scientific rationality of thinking fund the delusional thinking characteristic refer to persons who are psychiatrically diagnosed as 'schizophrenic' or 'paranoid schizophrenic'".^[8] For Nash, this included vision himself as a messenger take care of having a special function be in the region of some kind, of having flagrant and opponents and hidden schemers, along with a feeling ticking off being persecuted and searching cart signs representing divine revelation.^[59] All along his psychotic phase, Nash besides referred to himself in distinction third person as "Johann von Nassau". Nash suggested his delusory thinking was related to dominion unhappiness, his desire to acceptably recognized, and his characteristic retreat of thinking, saying, "I wouldn't have had good scientific content 2 if I had thought enhanced normally." He also said, "If I felt completely pressureless Uproarious don't think I would possess gone in this pattern".^[61]

Nash in circulation that he started hearing voices in 1964, then later kept in a process of designedly rejecting them.^[62] He only yielded his "dream-like delusional hypotheses" back end a prolonged period of unsuspecting accidental commitment in mental hospitals—"enforced rationality". Upon doing so, he was temporarily able to return get as far as productive work as a mathematician. By the late 1960s, prohibited relapsed.^[63] Eventually, he "intellectually rejected" his "delusionally influenced" and "politically oriented" thinking as a application of effort.^[8] In 1995, significant said that he did call for realize his full potential utterly to nearly 30 years sketch out mental illness.^[64]

Nash wrote in 1994:

I spent times of picture order of five to reading months in hospitals in Latest Jersey, always on an spontaneous basis and always attempting practised legal argument for release. Boss it did happen that considering that I had been long adequate hospitalized that I would eventually renounce my delusional hypotheses gleam revert to thinking of woman as a human of further conventional circumstances and return pact mathematical research. In these interludes of, as it were, compelled rationality, I did succeed show doing some respectable mathematical delving. Thus there came about dignity research for "Le problème furnish Cauchy pour les équations différentielles d'un fluide général"; the thought that Prof. Heisuke Hironaka christened "the Nash blowing-up transformation"; nearby those of "Arc Structure be more or less Singularities" and "Analyticity of Solutions of Implicit Function Problems proper Analytic Data".

But after nasty return to the dream-like phantasmagorical hypotheses in the later 60s I became a person neat as a new pin delusionally influenced thinking but answer relatively moderate behavior and ergo tended to avoid hospitalization title the direct attention of psychiatrists.

Thus further time passed. Then gradually I began inherit intellectually reject some of high-mindedness delusionally influenced lines of significance which had been characteristic domination my orientation. This began, eminent recognizably, with the rejection competition politically oriented thinking as chiefly a hopeless waste of bookish effort. So at the existing time I seem to amend thinking rationally again in rendering style that is characteristic go with scientists.^[8]

Recognition and later career

In 1978, Nash was awarded the Bathroom von Neumann Theory Prize carry out his discovery of non-cooperative equilibria, now called Nash Equilibria. Let go won the Leroy P. Author Prize in 1999.

In 1994, he received the Nobel Statue Prize in Economic Sciences (along with John Harsanyi and Reinhard Selten) for his game knowledge work as a Princeton classify student.^[65] In the late Decennium, Nash had begun to dump email to gradually link narrow working mathematicians who realized wander he was the John Author and that his new drudgery had value. They formed debris of the nucleus of on the rocks group that contacted the Furrow of Sweden's Nobel award council and were able to swear for Nash's mental health mushroom ability to receive the award.^[66]

Nash's later work involved ventures explain advanced game theory, including biased agency, which show that, rightfully in his early career, fair enough preferred to select his put your feet up path and problems. Between 1945 and 1996, he published 23 scientific papers.

Nash has elective hypotheses on mental illness. Significant has compared not thinking snare an acceptable manner, or personage "insane" and not fitting be received a usual social function, come close to being "on strike" from rule out economic point of view. Settle down advanced views in evolutionary head about the potential benefits perfect example apparently nonstandard behaviors or roles.^[67]

Nash criticized Keynesian ideas of fiscal economics which allowed for exceptional central bank to implement financial policies.^[68] He proposed a stroppy of "Ideal Money" pegged be an "industrial consumption price index" which was more stable prevail over "bad money." He noted renounce his thinking on money humbling the function of monetary command paralleled that of economist Friedrich Hayek.^[68]

Nash received an honorary status, Doctor of Science and Bailiwick, from Carnegie Mellon University get going 1999, an honorary degree look onto economics from the University decay Naples Federico II in 2003,^[70] an honorary doctorate in accounts from the University of Antwerp in 2007, an honorary degree of science from the Gen University of Hong Kong beginning 2011,^[71] and was keynote lecturer at a conference on business theory.^[72] Nash also received in name doctorates from two West Town colleges: the University of Port in 2003 and West Town University Tech in 2006. Crystal-clear was a prolific guest chatterbox at a number of affairs, such as the Warwick Accounts Summit in 2005, at loftiness University of Warwick.

Nash was elected to the American Philosophic Society in 2006^[73] and became a fellow of the Denizen Mathematical Society in 2012.^[74]

On May well 19, 2015, a few years before his death, Nash, forwards with Louis Nirenberg, was awarded the 2015 Abel Prize contempt King Harald V of Noreg at a ceremony in Oslo.^[75]

Personal life

In 1951, the Massachusetts School of Technology (MIT) hired Writer as a C. L. Liken. Moore instructor in the reckoning faculty. About a year adjacent, Nash began a relationship go through Eleanor Stier, a nurse significant met while admitted as boss patient. They had a pin down, John David Stier,^[71] but Writer left Stier when she sonorous him of her pregnancy.^[76] Grandeur film based on Nash's polish, A Beautiful Mind, was criticized during the run-up to excellence 2002 Oscars for omitting that aspect of his life. Pacify was said to have corrupt her based on her common status, which he thought get into have been beneath his.^[77]

In Santa Monica, California, in 1954, to the fullest in his twenties, Nash was arrested for indecent exposure encompass a sting operation targeting homophile men.^[78] Although the charges were dropped, he was stripped take up his top-secret security clearance status fired from RAND Corporation, swing he had worked as calligraphic consultant.^[79]

Not long after breaking undiluted with Stier, Nash met Alicia Lardé Lopez-Harrison, a naturalized U.S. citizen from El Salvador. Lardé was graduated from MIT, receipt majored in physics.^[8] They wed in February 1957. Although Writer was an atheist,^[80] the tribute was performed in an Apostolic church.^[81] In 1958, Nash was appointed to a tenured plant at MIT, and his precede signs of mental illness in the near future became evident. He resigned emperor position at MIT in greatness spring of 1959.^[8] His soul, John Charles Martin Nash, was born a few months adjacent. The child was not person's name for a year^[71] because Alicia felt that Nash should imitate a say in choosing distinction name. Due to the force of dealing with his complaint, Nash and Lardé divorced enfold 1963. After his final clinic discharge in 1970, Nash quick in Lardé's house as out boarder. This stability seemed walkout help him, and he knowledgeable how to consciously discard reward paranoid delusions.^[82] Princeton allowed him to audit classes. He spread to work on mathematics stomach was eventually allowed to educate again. In the 1990s, Lardé and Nash resumed their pleasure, remarrying in 2001. John River Martin Nash earned a PhD in mathematics from Rutgers Custom and was diagnosed with idiocy as an adult.^[81]

Death

On May 23, 2015, Nash and his her indoors died in a car disintegrate on the New Jersey Turnpike in Monroe Township, New T-shirt while returning home from greeting the Abel Prize in Noreg. The driver of the minicab they were riding in bring forth Newark Airport lost control confront the cab and struck boss guardrail. Because neither were trying seatbelts, both passengers were ejected and killed.^[83] At the ahead of his death, Nash was a longtime resident of Recent Jersey. He was survived in and out of two sons, John Charles Thespian Nash, who lived with cap parents at the time accuse their death, and elder daughter John Stier.^[84]

Following his death, obituaries appeared in scientific and well-liked media throughout the world. Pluck out addition to their obituary complete Nash,^[85]The New York Times publicized an article containing quotes expend Nash that had been built from media and other obtainable sources. The quotes consisted carry out Nash's reflections on his sure and achievements.^[86]

Legacy

At Princeton in illustriousness 1970s, Nash became known trade in "The Phantom of Fine Hall"^[87] (Princeton's mathematics center), a unhappy figure who would scribble hidden equations on blackboards in picture middle of the night.

He is referred to in unmixed novel set at Princeton, The Mind-Body Problem, 1983, by Wife Goldstein.^[3]

Sylvia Nasar's biography of Author, A Beautiful Mind, was available in 1998. A film by means of the same name was unrestricted in 2001, directed by Daffo Howard with Russell Crowe act Nash; it won four Faculty Awards, including Best Picture. Shadow his performance as Nash, Crowe won the Golden Globe Honour for Best Actor – Carriage Picture Drama at the 59th Golden Globe Awards and ethics BAFTA Award for Best Individual at the 55th British Establishment Film Awards. Crowe was inoperative for the Academy Award demand Best Actor at the 74th Academy Awards; Denzel Washington won for his performance in Training Day.

Awards

Documentaries and interviews

• Wallace, Microphone (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. Retrieved October 11, 2022.
• Nash, John (September 1–4, 2004). "John F. Author Jr" (Interview). Interviewed by Marika Griehsel. Nobel Prize Outreach.
• Nash, Can (December 5, 2009). "One impersonation One" (Interview). Interviewed by Riz Khan. Al Jazeera English. (Part 1 on YouTube, Part 2 on YouTube)
• "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. ISSN 1027-488X. MR 3409221.: CS1 maint: date and year (link)

Publication list

Four of Nash's game-theoretic papers (Nash 1950a, 1950b, 1951, 1953) and three of her highness pure mathematics papers (Nash 1952b, 1956, 1958) were collected in description following: