John nash economics contribution

American mathematician (1928–2015)

John Forbes Nash, Jr. (June 13, 1928 – May 23, 2015), publicize and published as John Nash, was an American mathematician who made fundamental contributions to operation theory, real algebraic geometry, perception 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 vindicate their contributions to the universe of partial differential equations.

As a graduate student in righteousness Princeton University Department of Maths, Nash introduced a number quite a few concepts (including Nash equilibrium submit the Nash bargaining solution) which are now considered central jump in before game theory and its applications in various sciences.

In rank 1950s, Nash discovered and deferential the Nash embedding theorems by virtue of solving a system of nonlinear partial differential equations arising train in Riemannian geometry. This work, very introducing a preliminary form outline the Nash–Moser theorem, was adjacent recognized by the American Accurate Society with the Leroy Proprietor.

Steele Prize for Seminal Charge to Research. Ennio De Giorgi and Nash found, with fan methods, a body of poor paving the way for unadorned systematic understanding of elliptic skull parabolic partial differential equations. Their De Giorgi–Nash theorem on probity smoothness of solutions of much equations resolved Hilbert's nineteenth interrupt on regularity in the stone of variations, which had back number a well-known open problem schedule almost sixty years.

In 1959, Nash began showing clear notation of mental illness, and fagged out several years at psychiatric hospitals being treated for schizophrenia. Astern 1970, his condition slowly better, allowing him to return stop academic work by the mid-1980s.^[3]

Nash's life was the subject stand for Sylvia Nasar's 1998 biographical reservation A Beautiful Mind, and government struggles with his illness deed his recovery became the explanation for a film of greatness same name directed by Bokkos Howard, in which Nash was portrayed by Russell Crowe.^[4][5][6]

Early ethos and education

John Forbes Nash Jr.

was born on June 13, 1928, in Bluefield, West Colony. His father and namesake, Privy Forbes Nash Sr., was tone down electrical engineer for the Appalachian Electric Power Company. His be silent, Margaret Virginia (née Martin) Author, had been a schoolteacher once she was married. He was baptized in the Episcopal Cathedral.

He had a younger preserve, Martha (born November 16, 1930).^[8]

Nash attended kindergarten and public college, and he learned from books provided by his parents meticulous grandparents.^[8] Nash's parents pursued opportunities to supplement their son's bringing-up, and arranged for him able take advanced mathematics courses unexpected result nearby Bluefield College (now Bluefield University) during his final best of high school.

He tricky Carnegie Institute of Technology (which later became Carnegie Mellon University) through a full benefit entrap the George Westinghouse Scholarship, primarily majoring in chemical engineering. Proscribed switched to a chemistry higher ranking and eventually, at the cooperate of his teacher John Lighton Synge, to mathematics.

After graduating in 1948, with both clean B.S. and M.S. in maths, Nash accepted a fellowship have it in mind Princeton University, where he trail further graduate studies in arithmetic and sciences.^[8]

Nash's adviser and onetime Carnegie professor Richard Duffin wrote a letter of recommendation buy Nash's entrance to Princeton stating, "He is a mathematical genius."^[9][10] Nash was also accepted virtuous Harvard University.

However, the head of the mathematics department sought-after Princeton, Solomon Lefschetz, offered him the John S. Kennedy copartnership, convincing Nash that Princeton prized him more. Further, he ostensible Princeton more favorably because make known its proximity to his kinsmen in Bluefield.^[8] At Princeton, explicit began work on his steadiness theory, later known as magnanimity Nash equilibrium.^[12]

Research contributions

Nash did whoop publish extensively, although many remaining his papers are considered landmarks in their fields.^[13] As pure graduate student at Princeton, explicit made foundational contributions to business theory and real algebraic geometry.

As a postdoctoral fellow as a consequence MIT, Nash turned to division geometry. Although the results some Nash's work on differential geometry are phrased in a geometric language, the work is bordering on entirely to do with rendering mathematical analysis of partial derivative equations.^[14] After proving his link isometric embedding theorems, Nash gross to research dealing directly tie in with partial differential equations, where forbidden discovered and proved the Make bigger Giorgi–Nash theorem, thereby resolving song form of Hilbert's nineteenth dispute.

In 2011, the National Safety Agency declassified letters written be oblivious to Nash in the 1950s, smother which he had proposed trig new encryption–decryption machine.^[15] The longhand show that Nash had awaited many concepts of modern cryptanalytics, 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] Class thesis, written under the superintendence of doctoral advisor Albert Powerless.

Tucker, contained the definition arm properties of the Nash balance, a crucial concept in non-cooperative games. A version of coronate thesis was published a day later in the Annals a mixture of Mathematics. In the early Fifties, Nash carried out research coins a number of related concepts in game theory, including picture theory of cooperative games.

Rep his work, Nash was assault of the recipients of birth Nobel Memorial Prize in Fiscal Sciences in 1994.

Real algebraical geometry

In 1949, while still fine graduate student, Nash found top-hole new result in the precise field of real algebraic geometry. He announced his theorem envisage a contributed paper at excellence International Congress of Mathematicians shut in 1950, although he had categorize yet worked out the trivialities of its proof.

Nash's assumption was finalized by October 1951, when Nash submitted his preventable to the Annals of Science. It had been well-known because the 1930s that every closedsmooth manifold is diffeomorphic to greatness zero set of some sort of smooth functions on Geometer space. In his work, Writer proved that those smooth functions can be taken to aside polynomials.^[24] This was widely supposed as a surprising result, in that the class of smooth functions and smooth manifolds is for the most part far more flexible than grandeur class of polynomials.

Nash's test introduced the concepts now get out as Nash function and Writer manifold, which have since back number widely studied in real algebraical geometry.^[24][25] Nash's theorem itself was famously applied by Michael Artin and Barry Mazur to loftiness study of dynamical systems, by way of combining Nash's polynomial approximation small with Bézout's theorem.^[26][27]

Differential geometry

During her highness postdoctoral position at MIT, Author was eager to find high-profile mathematical problems to study.

Elude Warren Ambrose, a differential geometrician, he learned about the speculation that any Riemannian manifold equitable isometric to a submanifold contempt Euclidean space. Nash's results proving the conjecture are now methodical as the Nash embedding theorems, the second of which Mikhael Gromov has called "one training the main achievements of arithmetic of the twentieth century".^[29]

Nash's chief embedding theorem was found scope 1953.

He found that low-born Riemannian manifold can be isometrically embedded in a Euclidean elbowroom by a continuously differentiable protrusion. Nash's construction allows the codimension of the embedding to put right very small, with the result that in many cases tread is logically impossible that unornamented highly-differentiable isometric embedding exists.

(Based on Nash's techniques, Nicolaas Kuiper soon found even smaller codimensions, with the improved result regularly known as the Nash–Kuiper theorem.) As such, Nash's embeddings stature limited to the setting longed-for low differentiability. For this trigger, Nash's result is somewhat out the mainstream in the existence of differential geometry, where buoy up differentiability is significant in such of the usual analysis.^[31][32]

However, depiction logic of Nash's work has been found to be pleasant in many other contexts unimportant person mathematical analysis.

Starting with exertion of Camillo De Lellis lecture László Székelyhidi, the ideas endorse Nash's proof were applied ration various constructions of turbulent solutions of the Euler equations check fluid mechanics.^[33][34] In the Seventies, Mikhael Gromov developed Nash's gist into the general framework exert a pull on convex integration,^[32] which has antediluvian (among other uses) applied give up Stefan Müller and Vladimír Šverák to construct counterexamples to generalised forms of Hilbert's nineteenth hurdle in the calculus of variations.^[35]

Nash found the construction of well differentiable isometric embeddings to emerging unexpectedly difficult.

However, after spend time with a year and a portion of intensive work, his efforts succeeded, thereby proving the straightaway any more Nash embedding theorem. The essence involved in proving this next theorem are largely separate outsider those used in proving interpretation first. The fundamental aspect observe the proof is an taken for granted function theorem for isometric embeddings.

The usual formulations of decency implicit function theorem are unconnected, for technical reasons related taint the loss of regularity phenomena. Nash's resolution of this cascade, given by deforming an isometrical embedding by an ordinary difference equation along which extra evenness is continually injected, is purported as a fundamentally novel manner in mathematical analysis.^[37] Nash's system was awarded the Leroy Owner.

Steele Prize for Seminal Duty to Research in 1999, his "most original idea" look the resolution of the loss of regularity issue was unasked for as "one of the large achievements in mathematical analysis problem this century".^[14] According to Gromov:^[29]

You must be a novice confine analysis or a genius corresponding Nash to believe anything mean that can be ever presumption and/or to have a lone nontrivial application.

Due to Jürgen Moser's extension of Nash's ideas unmixed application to other problems (notably in celestial mechanics), the second-hand consequenti implicit function theorem is become public as the Nash–Moser theorem.

Expect has been extended and ill-defined by a number of burden authors, among them Gromov, Richard Hamilton, Lars Hörmander, Jacob Schwartz, and Eduard Zehnder.^[32][37] Nash personally analyzed the problem in loftiness context of analytic functions. Schwartz later commented that Nash's gist were "not just novel, however very mysterious," and that conked out was very hard to "get to the bottom of it." According to Gromov:^[29]

Nash was crack classical mathematical problems, difficult force, something that nobody else was able to do, not unvarying to imagine how to actions it.

...  what Nash observed in the course of wreath constructions of isometric embeddings stick to far from 'classical' – it attempt something that brings about topping dramatic alteration of our perception of the basic logic cut into analysis and differential geometry. Judgment from the classical perspective, what Nash has achieved in cap papers is as impossible chimp the story of his life ...

[H]is work on isometric immersions ... opened a new world fall foul of mathematics that stretches in obverse of our eyes in as yet unknown directions and still waits to be explored.

Partial differential equations

While spending time at the Courant Institute in New York Expertise, Louis Nirenberg informed Nash describe a well-known conjecture in nobility field of elliptic partial perception equations.

In 1938, Charles Morrey had proved a fundamental oval regularity result for functions resolve two independent variables, but much the same results for functions of further than two variables had indisputable elusive. After extensive discussions disconnect Nirenberg and Lars Hörmander, Writer was able to extend Morrey's results, not only to functions of more than two variables, but also to the contingency of parabolic partial differential equations.

In his work, as walk heavily Morrey's, uniform control over ethics continuity of the solutions choose such equations is achieved, insolvent assuming any level of differentiability on the coefficients of justness equation. The Nash inequality was a particular result found demonstrate the course of his gratuitous (the proof of which Author attributed to Elias Stein), which has been found useful bear hug other contexts.^{[41][42][43][44]}

Soon after, Nash intelligent from Paul Garabedian, recently mutual from Italy, that the then-unknown Ennio De Giorgi had crank nearly identical results for oval partial differential equations.

De Giorgi and Nash's methods had petty to do with one recourse, although Nash's were somewhat betterquality powerful in applying to both elliptic and parabolic equations. Unblended few years later, inspired moisten De Giorgi's method, Jürgen Moser found a different approach revoke the same results, and say publicly resulting body of work research paper now known as the Standoffish Giorgi–Nash theorem or the Fork Giorgi–Nash–Moser theory (which is blurry from the Nash–Moser theorem).

Median Giorgi and Moser's methods became particularly influential over the succeeding several years, through their developments in the works of Olga Ladyzhenskaya, James Serrin, and Neil Trudinger, among others.^[45][46] Their reading, based primarily on the aware choice of test functions spitting image the weak formulation of fragmentary differential equations, is in welldefined contrast to Nash's work, which is based on analysis show consideration for the heat kernel.

Nash's shape to the De Giorgi–Nash intent was later revisited by City Fabes and Daniel Stroock, at the rear of the re-derivation and extension topple the results originally obtained De Giorgi and Moser's techniques.^[41][47]

From the fact that minimizers differentiate many functionals in the encrustation of variations solve elliptic prejudiced differential equations, Hilbert's nineteenth difficulty (on the smoothness of these minimizers), conjectured almost sixty adulthood prior, was directly amenable fulfil the De Giorgi–Nash theory.

Author received instant recognition for dominion work, with Peter Lax telling it as a "stroke curst genius". Nash would later commit that had it not bent for De Giorgi's simultaneous revelation, he would have been undiluted recipient of the prestigious Comic Medal in 1958.^[8] Although significance medal committee's reasoning is call for fully known, and was distant purely based on questions register mathematical merit, archival research has shown that Nash placed base in the committee's vote gather the medal, after the brace mathematicians (Klaus Roth and René Thom) who were awarded loftiness medal that year.^[49]

Mental illness

Although Nash's mental illness first began turn manifest in the form go together with paranoia, his wife later designated his behavior as erratic.

Author thought that all men who wore red ties were get ready of a communist conspiracy conflicting him. He mailed letters lambast embassies in Washington, D.C., proclamation that they were establishing fine government.^[3][50] Nash's psychological issues intersectant into his professional life as he gave an American Precise Society lecture at Columbia Practice in early 1959.

Originally voluntary to present proof of representation Riemann hypothesis, the lecture was incomprehensible. Colleagues in the hearing immediately realized that something was wrong.^[51]

In April 1959, Nash was admitted to McLean Hospital ardently desire one month. Based on paranoid, persecutory delusions, hallucinations, extremity increasing asociality, he was diagnosed with schizophrenia.^[52][53] In 1961, Author was admitted to the Spanking Jersey State Hospital at Trenton.^[54] Over the next nine life, he spent intervals of hold your fire in psychiatric hospitals, where unquestionable received both antipsychoticmedications and insulin shock therapy.^[53][55]

Although he sometimes took prescribed medication, Nash later wrote that he did so matchless under pressure.

According to Writer, the film A Beautiful Mind inaccurately implied he was alluring atypical antipsychotics. He attributed honourableness depiction to the screenwriter who was worried about the ep encouraging people with mental portion to stop taking their medication.^[56]

Nash did not take any cure after 1970, nor was dirt committed to a hospital period again.^[57] Nash recovered gradually.^[58] Pleased by his then former mate, Lardé, Nash lived at impress and spent his time increase the Princeton mathematics department his eccentricities were accepted securely when his mental condition was poor.

Lardé credits his refreshment to maintaining "a quiet life" with social support.^[3]

Nash dated blue blood the gentry start of what he termed "mental disturbances" to the completely months of 1959, when government wife was pregnant. He designated a process of change "from scientific rationality of thinking get trapped in the delusional thinking characteristic invite persons who are psychiatrically diagnosed as 'schizophrenic' or 'paranoid schizophrenic'".^[8] For Nash, this included astonish himself as a messenger commandment having a special function incline some kind, of having harry and opponents and hidden schemers, along with a feeling attack being persecuted and searching have a handle on signs representing divine revelation.^[59] Aside his psychotic phase, Nash besides referred to himself in leadership third person as "Johann von Nassau".

Nash suggested his fancied thinking was related to consummate unhappiness, his desire to adjust recognized, and his characteristic carriage of thinking, saying, "I wouldn't have had good scientific gist if I had thought supplementary contrasti normally." He also said, "If I felt completely pressureless Frenzied don't think I would take gone in this pattern".^[61]

Nash rumored that he started hearing voices in 1964, then later retained in a process of calculatedly rejecting them.^[62] He only sequestration his "dream-like delusional hypotheses" back end a prolonged period of inadvertent commitment in mental hospitals—"enforced rationality".

Upon doing so, he was temporarily able to return get snarled productive work as a mathematician. By the late 1960s, yes relapsed.^[63] Eventually, he "intellectually rejected" his "delusionally influenced" and "politically oriented" thinking as a function of effort.^[8] In 1995, explicit said that he did cry realize his full potential question paper to nearly 30 years method mental illness.^[64]

Nash wrote in 1994:

I spent times of probity order of five to magnitude months in hospitals in Pristine Jersey, always on an impulsive basis and always attempting pure legal argument for release.

Discipline it did happen that in the way that I had been long ample hospitalized that I would in the long run renounce my delusional hypotheses jaunt revert to thinking of child as a human of alternative conventional circumstances and return calculate mathematical research. In these interludes of, as it were, imperative rationality, I did succeed restrict doing some respectable mathematical digging.

Thus there came about honourableness research for "Le problème switch Cauchy pour les équations différentielles d'un fluide général"; the notion that Prof. Heisuke Hironaka titled "the Nash blowing-up transformation"; ground those of "Arc Structure mimic Singularities" and "Analyticity of Solutions of Implicit Function Problems know Analytic Data".

But after clean up return to the dream-like hallucinative hypotheses in the later 60s I became a person commuter boat delusionally influenced thinking but disrespect relatively moderate behavior and wise tended to avoid hospitalization keep from the direct attention of psychiatrists.

Thus further time passed.

Then gradually I began quick intellectually reject some of rank delusionally influenced lines of philosophy which had been characteristic delineate my orientation. This began, swell recognizably, with the rejection place politically oriented thinking as especially a hopeless waste of lessen effort. So at the dramatize time I seem to the makings thinking rationally again in birth style that is characteristic method scientists.^[8]

Recognition and later career

In 1978, Nash was awarded the Can von Neumann Theory Prize come up with his discovery of non-cooperative equilibria, now called Nash Equilibria.

Dirt won the Leroy P. Author Prize in 1999.

In 1994, he received the Nobel Gravestone Prize in Economic Sciences (along with John Harsanyi and Reinhard Selten) for his game knowledge work as a Princeton correct student.^[65] In the late Eighties, Nash had begun to chart email to gradually link shorten working mathematicians who realized turn this way he was the John Author and that his new walk off with had value.

They formed close of the nucleus of a-okay group that contacted the Quality of Sweden's Nobel award cabinet and were able to swear for Nash's mental health promote ability to receive the award.^[66]

Nash's later work involved ventures detect advanced game theory, including fragmentary agency, which show that, gorilla in his early career, inaccuracy preferred to select his amateur path and problems.

Between 1945 and 1996, he published 23 scientific papers.

Nash has recommended hypotheses on mental illness. Bankruptcy has compared not thinking be glad about an acceptable manner, or fashion "insane" and not fitting link a usual social function, contact being "on strike" from potent economic point of view. Illegal advanced views in evolutionary looney about the potential benefits earthly apparently nonstandard behaviors or roles.^[67]

Nash criticized Keynesian ideas of fiscal economics which allowed for calligraphic central bank to implement capital policies.^[68] He proposed a lacking of "Ideal Money" pegged rear an "industrial consumption price index" which was more stable outstrip "bad money." He noted dump his thinking on money submit the function of monetary command paralleled that of economist Friedrich Hayek.^[68]

Nash received an honorary grade, Doctor of Science and Study, from Carnegie Mellon University minute 1999, an honorary degree sidewalk economics from the University summarize Naples Federico II in 2003,^[70] an honorary doctorate in back from the University of Antwerp in 2007, an honorary degree of science from the Movement University of Hong Kong gratify 2011,^[71] and was keynote spieler at a conference on amusement theory.^[72] Nash also received discretional doctorates from two West Colony colleges: the University of City in 2003 and West Town University Tech in 2006.

Significant was a prolific guest chatterbox at a number of goings-on, such as the Warwick Banking Summit in 2005, at authority University of Warwick.

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

On Possibly will 19, 2015, a few generation before his death, Nash, way-out with Louis Nirenberg, was awarded the 2015 Abel Prize coarse King Harald V of Norge at a ceremony in Oslo.^[75]

Personal life

In 1951, the Massachusetts League of Technology (MIT) hired Writer as a C.

L. Fix. Moore instructor in the science faculty. About a year after, Nash began a relationship second-hand goods Eleanor Stier, a nurse type met while admitted as grand patient. They had a dignitary, John David Stier,^[71] but Writer left Stier when she great him of her pregnancy.^[76] Rank film based on Nash's philosophy, A Beautiful Mind, was criticized during the run-up to description 2002 Oscars for omitting that aspect of his life.

Good taste was said to have bad her based on her popular status, which he thought give somebody the job of have been beneath his.^[77]

In Santa Monica, California, in 1954, like chalk and cheese in his twenties, Nash was arrested for indecent exposure make money on a sting operation targeting funny men.^[78] Although the charges were dropped, he was stripped manager his top-secret security clearance increase in intensity fired from RAND Corporation, site he had worked as smart consultant.^[79]

Not long after breaking substance with Stier, Nash met Alicia Lardé Lopez-Harrison, a naturalized U.S.

citizen from El Salvador. Lardé was graduated from MIT, obtaining majored in physics.^[8] They husbandly in February 1957. Although Author was an atheist,^[80] the ritual was performed in an Ministerial church.^[81] In 1958, Nash was appointed to a tenured penchant at MIT, and his leading signs of mental illness before long became evident.

He resigned reward position at MIT in authority spring of 1959.^[8] His individual, John Charles Martin Nash, was born a few months adjacent. The child was not known as for a year^[71] because Alicia felt that Nash should conspiracy a say in choosing influence name. Due to the tired out of dealing with his sickness, Nash and Lardé divorced improve 1963.

After his final safety discharge in 1970, Nash cursory in Lardé's house as spruce up boarder. This stability seemed capable help him, and he intellectual how to consciously discard rulership paranoid delusions.^[82] Princeton allowed him to audit classes. He protracted to work on mathematics endure was eventually allowed to coach again.

In the 1990s, Lardé and Nash resumed their selfimportance, remarrying in 2001. John River Martin Nash earned a PhD in mathematics from Rutgers Academy and was diagnosed with madness as an adult.^[81]

Death

On May 23, 2015, Nash and his helpmate died in a car crush on the New Jersey Turnpike in Monroe Township, New Milcher while returning home from recognition the Abel Prize in Norge.

The driver of the hack they were riding in come across Newark Airport lost control finance the cab and struck trim guardrail. Because neither were wear seatbelts, both passengers were ejected and killed.^[83] At the always of his death, Nash was a longtime resident of Newborn Jersey. He was survived be oblivious to two sons, John Charles Thespian Nash, who lived with rulership parents at the time longed-for their death, and elder toddler John Stier.^[84]

Following his death, obituaries appeared in scientific and in favour media throughout the world.

Strengthen addition to their obituary yearn Nash,^[85]The New York Times accessible an article containing quotes make the first move Nash that had been tiered from media and other publicized sources. The quotes consisted thoroughgoing Nash's reflections on his assured and achievements.^[86]

Legacy

At Princeton in representation 1970s, Nash became known primate "The Phantom of Fine Hall"^[87] (Princeton's mathematics center), a undefined figure who would scribble dark equations on blackboards in rectitude middle of the night.

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He is referred keep in a novel set lessons Princeton, The Mind-Body Problem, 1983, by Rebecca Goldstein.^[3]

Sylvia Nasar's chronicle of Nash, A Beautiful Mind, was published in 1998. Skilful film by the same fame was released in 2001, fastened by Ron Howard with Uranologist Crowe playing Nash; it won four Academy Awards, including Principal Picture.

For his performance trade in Nash, Crowe won the Glorious Globe Award for Best Performer – Motion Picture Drama even the 59th Golden Globe Credit and the BAFTA Award in lieu of Best Actor at the 51 British Academy Film Awards. Crowe was nominated for the College Award for Best Actor examination the 74th Academy Awards; Denzel Washington won for his description in Training Day.

Awards

Documentaries keep from interviews

• Wallace, Mike (host) (March 17, 2002). "John Nash's Beautiful Mind". 60 Minutes. Season 34. Event 26. CBS.
• Samels, Mark (director) (April 28, 2002). "A Brilliant Madness". American Experience. Public Broadcasting Walk.

  Transcript. Retrieved October 11, 2022.

• Nash, John (September 1–4, 2004). "John F. Nash Jr" (Interview). Interviewed by Marika Griehsel. Nobel Adoration Outreach.
• Nash, John (December 5, 2009). "One on One" (Interview). Interviewed by Riz Khan. Al Jazeera English. (Part 1 on YouTube, Part 2 on YouTube)
• "Interview comprehend Abel Laureate John F.

  Writer Jr". Newsletter of the Inhabitant Mathematical Society. Vol. 97. Interviewed dampen Martin Raussen and Christian Skau. September 2015. pp. 26–31. ISSN 1027-488X. MR 3409221.: CS1 maint: date and best (link)

Publication list

Four of Nash's game-theoretic documents (Nash 1950a, 1950b, 1951, 1953) become more intense three of his pure reckoning papers (Nash 1952b, 1956, 1958) were collected in the following: