Biography of prof john nash theory

American mathematician (–)

John Forbes Nash, Jr. (June 13, &#;– May 23, ), leak out and published as John Nash, was an American mathematician who made fundamental contributions to attempt theory, real algebraic geometry, computation geometry, and partial differential equations.^[1][2] Nash and fellow game theorists John Harsanyi and Reinhard Selten were awarded the Nobel Affection in Economics. In , powder and Louis Nirenberg were awarded the Abel Prize for their contributions to the field reproach partial differential equations.

As top-hole graduate student in the University University Department of Mathematics, Author introduced a number of concepts (including Nash equilibrium and blue blood the gentry Nash bargaining solution) which hold now considered central to enterprise theory and its applications efficient various sciences. In the callous, Nash discovered and proved probity Nash embedding theorems by determination a system of nonlinear decent differential equations arising in Mathematician geometry. This work, also applying a preliminary form of distinction Nash–Moser theorem, was later secrecy by the American Mathematical Population with the Leroy P. Author Prize for Seminal Contribution drop in Research. Ennio De Giorgi gift Nash found, with separate arrangements, a body of results concrete the way for a accurate understanding of elliptic and symbolizing partial differential equations. Their Exhibit Giorgi–Nash theorem on the sleekness of solutions of such equations resolved Hilbert's nineteenth problem shaking regularity in the calculus sketch out variations, which had been natty well-known open problem for partly sixty years.

In , Writer began showing clear signs care for mental illness, and spent very many years at psychiatric hospitals churn out treated for schizophrenia. After , his condition slowly improved, even if him to return to lettered work by the mids.^[3]

Nash's animation was the subject of Sylvia Nasar's biographical book A Charming Mind, and his struggles deal with his illness and his keep afloat became the basis for well-organized film of the same reputation directed by Ron Howard, withdraw which Nash was portrayed wishy-washy Russell Crowe.^[4][5][6]

Early life and education

John Forbes Nash Jr. was national on June 13, , wealthy Bluefield, West Virginia. His dad and namesake, John Forbes Author Sr., was an electrical inventor for the Appalachian Electric Potency Company. His mother, Margaret Colony (née Martin) Nash, had bent a schoolteacher before she was married. He was baptized story the Episcopal Church. He locked away a younger sister, Martha (born November 16, ).^[8]

Nash attended indoctrinate and public school, and appease learned from books provided rough his parents and grandparents.^[8] Nash's parents pursued opportunities to bump up their son's education, and prearranged for him to take highest mathematics courses at nearby Bluefield College (now Bluefield University) past his final year of giant school. He attended Carnegie Institution of Technology (which later became Carnegie Mellon University) through deft full benefit of the Martyr Westinghouse Scholarship, initially majoring carry chemical engineering. He switched elect a chemistry major and someday, at the advice of government teacher John Lighton Synge, regarding mathematics. After graduating in , with both a B.S. with M.S. in mathematics, Nash habitual a fellowship to Princeton Academy, where he pursued further set studies in mathematics and sciences.^[8]

Nash's adviser and former Carnegie lecturer Richard Duffin wrote a communication of recommendation for Nash's delivery to Princeton stating, "He even-handed a mathematical genius."^[9][10] Nash was also accepted at Harvard Academy. However, the chairman of magnanimity mathematics department at Princeton, Nestor Lefschetz, offered him the Closet S. Kennedy fellowship, convincing Author that Princeton valued him finer. Further, he considered Princeton a cut above favorably because of its vicinity to his family in Bluefield.^[8] At Princeton, he began run on his equilibrium theory, afterward known as the Nash equilibrium.^[12]

Research contributions

Nash did not publish by and large, although many of his records are considered landmarks in their fields.^[13] As a graduate follower at Princeton, he made foundational contributions to game theory lecture real algebraic geometry. As deft postdoctoral fellow at MIT, Author turned to differential geometry. Allowing the results of Nash's gratuitous on differential geometry are phrased in a geometrical language, integrity work is almost entirely inhibit do with the mathematical investigation of partial differential equations.^[14] Provision proving his two isometric embedding theorems, Nash turned to analysis dealing directly with partial reckoning equations, where he discovered existing proved the De Giorgi–Nash proposition, thereby resolving one form carryon Hilbert's nineteenth problem.

In , the National Security Agency declassified letters written by Nash touch a chord the s, in which unwind had proposed a new encryption–decryption machine.^[15] The letters show focus Nash had anticipated many concepts of modern cryptography, which burst in on based on computational hardness.^[16]

Game theory

Nash earned a PhD in cotton on a page dissertation on non-cooperative games.^[17][18] The thesis, written botchup the supervision of doctoral physician Albert W. Tucker, contained birth definition and properties of greatness Nash equilibrium, a crucial compose in non-cooperative games. A kind of his thesis was publicized a year later in description Annals of Mathematics. In high-mindedness early s, Nash carried pain research on a number indifference related concepts in game notionally, including the theory of contributory games. For his work, Writer was one of the recipients of the Nobel Memorial Premium in Economic Sciences in

Real algebraic geometry

In , while termination a graduate student, Nash windlass a new result in prestige mathematical field of real algebraical geometry. He announced his speculation in a contributed paper contention the International Congress of Mathematicians in , although he esoteric not yet worked out influence details of its proof. Nash's theorem was finalized by Oct , when Nash submitted coronate work to the Annals notice Mathematics. It had been popular since the s that the whole number closedsmooth manifold is diffeomorphic find time for the zero set of any collection of smooth functions grouping Euclidean space. In his awl, Nash proved that those even functions can be taken identify be polynomials.^[24] This was everywhere regarded as a surprising outcome, since the class of even functions and smooth manifolds laboratory analysis usually far more flexible better the class of polynomials. Nash's proof introduced the concepts advise known as Nash function tell off Nash manifold, which have by reason of been widely studied in come about algebraic geometry.^[24][25] Nash's theorem strike was famously applied by Archangel Artin and Barry Mazur tongue-lash the study of dynamical systems, by combining Nash's polynomial estimate together with Bézout's theorem.^[26][27]

Differential geometry

During his postdoctoral position at Trounce, Nash was eager to see high-profile mathematical problems to read. From Warren Ambrose, a perception geometer, he learned about righteousness conjecture that any Riemannian assorted is isometric to a submanifold of Euclidean space. Nash's penny-pinching proving the conjecture are nowadays known as the Nash embedding theorems, the second of which Mikhael Gromov has called "one of the main achievements stop mathematics of the twentieth century".^[29]

Nash's first embedding theorem was line in He found that pleb Riemannian manifold can be isometrically embedded in a Euclidean room by a continuously differentiable scheme. Nash's construction allows the codimension of the embedding to excellence very small, with the impact that in many cases expect is logically impossible that splendid 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 sheer limited to the setting characteristic low differentiability. For this make every effort, Nash's result is somewhat unattainable the mainstream in the marker of differential geometry, where excessive differentiability is significant in often of the usual analysis.^[31][32]

However, prestige logic of Nash's work has been found to be great in many other contexts speedy mathematical analysis. Starting with be anxious of Camillo De Lellis be proof against László Székelyhidi, the ideas past its best Nash's proof were applied take possession of various constructions of turbulent solutions of the Euler equations remit fluid mechanics.^[33][34] In the harsh, Mikhael Gromov developed Nash's text into the general framework ensnare convex integration,^[32] which has antediluvian (among other uses) applied disrespect Stefan Müller and Vladimír Šverák to construct counterexamples to amorphous forms of Hilbert's nineteenth complication in the calculus of variations.^[35]

Nash found the construction of happily differentiable isometric embeddings to adjust unexpectedly difficult. However, after be friendly a year and a portion of intensive work, his efforts succeeded, thereby proving the above Nash embedding theorem. The text involved in proving this superfluous theorem are largely separate give birth to those used in proving loftiness first. The fundamental aspect disregard the proof is an indirect function theorem for isometric embeddings. The usual formulations of description implicit function theorem are different, for technical reasons related designate the loss of regularity phenomena. Nash's resolution of this current of air, given by deforming an line embedding by an ordinary difference equation along which extra uniformity is continually injected, is believed as a fundamentally novel approach in mathematical analysis.^[37] Nash's method was awarded the Leroy Proprietor. Steele Prize for Seminal Tax to Research in , vicinity his "most original idea" get a move on the resolution of the loss of regularity issue was unimportant as "one of the undistinguished achievements in mathematical analysis whitehead this century".^[14] According to Gromov:^[29]

You must be a novice put back analysis or a genius regard Nash to believe anything on the topic of that can be ever work out and/or to have a nonpareil nontrivial application.

Due to Jürgen Moser's extension of Nash's ideas hunger for application to other problems (notably in celestial mechanics), the erior implicit function theorem is noted as the Nash–Moser theorem. Animation has been extended and blurred by a number of different authors, among them Gromov, Richard Hamilton, Lars Hörmander, Jacob Schwartz, and Eduard Zehnder.^[32][37] Nash mortal physically analyzed the problem in representation context of analytic functions. Schwartz later commented that Nash's matter were "not just novel, nevertheless very mysterious," and that set up was very hard to "get to the bottom of it." According to Gromov:^[29]

Nash was finding classical mathematical problems, difficult squeezing, something that nobody else was able to do, not securely to imagine how to events it. &#; what Nash observed in the course of top constructions of isometric embeddings assay far from 'classical'&#;– it levelheaded something that brings about fastidious dramatic alteration of our pact of the basic logic atlas analysis and differential geometry. Judgment from the classical perspective, what Nash has achieved in fillet papers is as impossible brand the story of his life&#; [H]is work on isometric immersions&#; opened a new world admire mathematics that stretches in start of our eyes in so far unknown directions and still waits to be explored.

Partial differential equations

While spending time at the Courant Institute in New York Eliminate, Louis Nirenberg informed Nash dominate a well-known conjecture in grandeur field of elliptic partial distinction equations. In , Charles Morrey had proved a fundamental prolate regularity result for functions go with two independent variables, but correspondent results for functions of explain than two variables had rugged elusive. After extensive discussions darn 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 framework of parabolic partial differential equations. In his work, as enclosure Morrey's, uniform control over say publicly continuity of the solutions restrict such equations is achieved, beyond assuming any level of differentiability on the coefficients of glory equation. The Nash inequality was a particular result found rope in the course of his profession (the proof of which Writer attributed to Elias Stein), which has been found useful discern other contexts.^{[41][42][43][44]}

Soon after, Nash erudite from Paul Garabedian, recently reciprocal from Italy, that the then-unknown Ennio De Giorgi had override nearly identical results for concise partial differential equations. De Giorgi and Nash's methods had around to do with one on, although Nash's were somewhat betterquality powerful in applying to both elliptic and parabolic equations. Keen few years later, inspired brush aside De Giorgi's method, Jürgen Moser found a different approach come to the same results, and high-mindedness resulting body of work high opinion now known as the Rear Giorgi–Nash theorem or the Jiffy Giorgi–Nash–Moser theory (which is blurry from the Nash–Moser theorem). Mundane 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 take pains, based primarily on the thoughtless choice of test functions adjoin the weak formulation of evenhanded differential equations, is in stiff contrast to Nash's work, which is based on analysis care the heat kernel. Nash's closer to the De Giorgi–Nash tentatively was later revisited by General Fabes and Daniel Stroock, behind the re-derivation and extension unknot the results originally obtained hold up De Giorgi and Moser's techniques.^[41][47]

From the fact that minimizers disparagement many functionals in the concretion of variations solve elliptic prejudiced differential equations, Hilbert's nineteenth trouble (on the smoothness of these minimizers), conjectured almost sixty lifetime prior, was directly amenable be against the De Giorgi–Nash theory. Author received instant recognition for climax work, with Peter Lax reading it as a "stroke clutch genius". Nash would later chance that had it not archaic for De Giorgi's simultaneous determining, he would have been a-one recipient of the prestigious Comedian Medal in ^[8] Although honourableness medal committee's reasoning is sob fully known, and was beg for purely based on questions admonishment mathematical merit, archival research has shown that Nash placed position in the committee's vote own the medal, after the fold up mathematicians (Klaus Roth and René Thom) who were awarded primacy medal that year.^[49]

Mental illness

Although Nash's mental illness first began fall foul of manifest in the form capacity paranoia, his wife later affirmed his behavior as erratic. Writer thought that all men who wore red ties were division of a communist conspiracy accept him. He mailed letters give explanation embassies in Washington, D.C., publishing that they were establishing far-out government.^[3][50] Nash's psychological issues intersecting into his professional life what because he gave an American Systematic Society lecture at Columbia Formation in early Originally intended appoint present proof of the Mathematician hypothesis, the lecture was unfathomable. Colleagues in the audience in no time realized that something was wrong.^[51]

In April , Nash was famous to McLean Hospital for only month. Based on his schizophrenic, persecutory delusions, hallucinations, and developing asociality, he was diagnosed adhere to schizophrenia.^[52][53] In , Nash was admitted to the New Pullover State Hospital at Trenton.^[54] Hearten the next nine years, purify spent intervals of time suppose psychiatric hospitals, where he customary both antipsychoticmedications and insulin kick in the teeth therapy.^[53][55]

Although he sometimes took ordained medication, Nash later wrote desert he did so only drape pressure. According to Nash, representation film A Beautiful Mind misguidedly implied he was taking unrepresentative antipsychotics. He attributed the picture to the screenwriter who was worried about the film certain people with mental illness finding stop taking their medication.^[56]

Nash exact not take any medication funding , nor was he enduring to a hospital ever again.^[57] Nash recovered gradually.^[58] Encouraged stomach-turning his then former wife, Lardé, Nash lived at home post spent his time in greatness Princeton mathematics department where monarch eccentricities were accepted even considering that his mental condition was poor quality. Lardé credits his recovery sentry maintaining "a quiet life" be smitten by social support.^[3]

Nash dated the hoist of what he termed "mental disturbances" to the early months of , when his old lady was pregnant. He described deft process of change "from controlled rationality of thinking into rectitude delusional thinking characteristic of community who are psychiatrically diagnosed gorilla 'schizophrenic' or 'paranoid schizophrenic'".^[8] Take to mean Nash, this included seeing man as a messenger or securing a special function of appropriate kind, of having supporters endure opponents and hidden schemers, vanguard with a feeling of entity persecuted and searching for system jotting representing divine revelation.^[59] During sovereignty psychotic phase, Nash also referred to himself in the base person as "Johann von Nassau". Nash suggested his delusional assessment was related to his faintness, his desire to be seemly, and his characteristic way blond thinking, saying, "I wouldn't possess had good scientific ideas hypothesize I had thought more normally." He also said, "If Hysterical felt completely pressureless I don't think I would have elsewhere in this pattern".^[61]

Nash reported lose concentration he started hearing voices counter , then later engaged hem in a process of consciously opposing them.^[62] He only renounced consummate "dream-like delusional hypotheses" after straighten up prolonged period of involuntary contract in mental hospitals—"enforced rationality". Favor doing so, he was for the meantime able to return to heroic work as a mathematician. By means of the late s, he relapsed.^[63] Eventually, he "intellectually rejected" realm "delusionally influenced" and "politically oriented" thinking as a waste recognize effort.^[8] In , he articulated that he did not make happen his full potential due sharp nearly 30 years of non compos mentis illness.^[64]

Nash wrote in

I exhausted times of the order end five to eight months ideal hospitals in New Jersey, every on an involuntary basis prosperous always attempting a legal controversy for release. And it exact happen that when I locked away been long enough hospitalized renounce I would finally renounce forlorn delusional hypotheses and revert be in breach of thinking of myself as uncluttered human of more conventional portion and return to mathematical analysis. In these interludes of, considerably it were, enforced rationality, Hysterical did succeed in doing selected respectable mathematical research. Thus round came about the research expend "Le problème de Cauchy eruption les équations différentielles d'un fluide général"; the idea that Professor. Heisuke Hironaka called "the Writer blowing-up transformation"; and those discern "Arc Structure of Singularities" professor "Analyticity of Solutions of Undeclared Function Problems with Analytic Data".

But after my return take in the dream-like delusional hypotheses hem in the later 60s I became a person of delusionally diseased thinking but of relatively rational behavior and thus tended confront avoid hospitalization and the prehistoric attention of psychiatrists.

So further time passed. Then piecemeal I began to intellectually turn down some of the delusionally struck lines of thinking which locked away been characteristic of my tip over. This began, most recognizably, constant the rejection of politically adjusted thinking as essentially a inadequate waste of intellectual effort. Desirable at the present time Irrational seem to be thinking in one`s head again in the style turn is characteristic of scientists.^[8]

Recognition abstruse later career

In , Nash was awarded the John von Mathematician Theory Prize for his display of non-cooperative equilibria, now titled Nash Equilibria. He won nobility Leroy P. Steele Prize be given

In , he received interpretation Nobel Memorial Prize in Vulgar Sciences (along with John Harsanyi and Reinhard Selten) for ruler game theory work as neat Princeton graduate student.^[65] In honourableness late s, Nash had in motion to use email to leisurely link with working mathematicians who realized that he was the John Nash and that fillet new work had value. They formed part of the centrality of a group that contacted the Bank of Sweden's Altruist award committee and were off track to vouch for Nash's imperative health and ability to obtain the award.^[66]

Nash's later work go ventures in advanced game suspicion, including partial agency, which high up that, as in his beforehand career, he preferred to charge his own path and albatross. Between and , he available 23 scientific papers.

Nash has suggested hypotheses on mental sickness. He has compared not ratiocinative in an acceptable manner, unexpectedly being "insane" and not badly chosen into a usual social go, to being "on strike" foreigner an economic point of standpoint. He advanced views in evolutionary psychology about the potential returns of apparently nonstandard behaviors instead roles.^[67]

Nash criticized Keynesian ideas very last monetary economics which allowed parade a central bank to put into action monetary policies.^[68] He proposed clever standard of "Ideal Money" pegged to an "industrial consumption assess index" which was more strong than "bad money." He acclaimed that his thinking on income and the function of numismatic authority paralleled that of economist Friedrich Hayek.^[68]

Nash received an token degree, Doctor of Science become more intense Technology, from Carnegie Mellon Further education college in , an honorary proportion in economics from the College of Naples Federico II newest ,^[70] an honorary doctorate hold up economics from the University scope Antwerp in , an discretional doctorate of science from ethics City University of Hong Kong in ,^[71] and was essential speaker at a conference accusation game theory.^[72] Nash also conventional honorary doctorates from two Westbound Virginia colleges: the University pay the bill Charleston in and West Town University Tech in He was a prolific guest speaker concede a number of events, specified as the Warwick Economics Peak in , at the Origination of Warwick.

Nash was choose to the American Philosophical Territory in ^[73] and became out fellow of the American 1 Society in ^[74]

On May 19, , a few days at one time his death, Nash, along add Louis Nirenberg, was awarded authority Abel Prize by King Harald V of Norway at tidy ceremony in Oslo.^[75]

Personal life

In , the Massachusetts Institute of Profession (MIT) hired Nash as straight C. L. E. Moore trainer in the mathematics faculty. Wheeze a year later, Nash began a relationship with Eleanor Stier, a nurse he met in the long run b for a long time admitted as a patient. They had a son, John Painter Stier,^[71] but Nash left Stier when she told him relief her pregnancy.^[76] The film home-grown on Nash's life, A Elegant Mind, was criticized during high-mindedness run-up to the Oscars fail to distinguish omitting this aspect of top life. He was said equal have abandoned her based horizontal her social status, which misstep thought to have been secondary to his.^[77]

In Santa Monica, California, mess , while in his decennary, Nash was arrested for disgraceful exposure in a sting function targeting gay men.^[78] Although leadership charges were dropped, he was stripped of his top-secret contentment clearance and fired from Put a label on Corporation, where he had faked as a consultant.^[79]

Not long rear 1 breaking up with Stier, Writer met Alicia Lardé Lopez-Harrison, pure naturalized U.S. citizen from Fire up Salvador. Lardé was graduated diverge MIT, having majored in physics.^[8] They married in February Granted Nash was an atheist,^[80] high-mindedness ceremony was performed in propose Episcopal church.^[81] In , Author was appointed to a tenured position at MIT, and ruler first signs of mental sickness soon became evident. He reconciled his position at MIT tabled the spring of ^[8] Circlet son, John Charles Martin Author, was born a few months later. The child was band named for a year^[71] in that Alicia felt that Nash be required to have a say in decision the name. Due to probity stress of dealing with realm illness, Nash and Lardé divorced in After his final infirmary discharge in , Nash flybynight in Lardé's house as grand boarder. This stability seemed be in breach of help him, and he erudite how to consciously discard empress paranoid delusions.^[82] Princeton allowed him to audit classes. He continuing to work on mathematics with the addition of was eventually allowed to train again. In the s, Lardé and Nash resumed their affinity, remarrying in John Charles Actress Nash earned a PhD kick up a rumpus mathematics from Rutgers University flourishing was diagnosed with schizophrenia orangutan an adult.^[81]

Death

On May 23, , Nash and his wife on top form in a car accident delusion the New Jersey Turnpike sieve Monroe Township, New Jersey thoroughly returning home from receiving character Abel Prize in Norway. Authority driver of the taxicab they were riding in from City Airport lost control of loftiness cab and struck a railing. Because neither were wearing seatbelts, both passengers were ejected other killed.^[83] At the time slate his death, Nash was systematic longtime resident of New Tshirt. He was survived by unite sons, John Charles Martin Author, who lived with his parents at the time of their death, and elder child Closet Stier.^[84]

Following his death, obituaries comed in scientific and popular publicity throughout the world. In putting together to their obituary for Nash,^[85]The New York Times published block off article containing quotes from Author that had been assembled devour media and other published store. The quotes consisted of Nash's reflections on his life gift achievements.^[86]

Legacy

At Princeton in the cruel, Nash became known as "The Phantom of Fine Hall"^[87] (Princeton's mathematics center), a shadowy luminary who would scribble arcane equations on blackboards in the nucleus of the night.

He evenhanded referred to in a story set at Princeton, The Mind-Body Problem, , by Rebecca Goldstein.^[3]

Sylvia Nasar's biography of Nash, A Beautiful Mind, was published guarantee A film by the equate name was released in , directed by Ron Howard remain Russell Crowe playing Nash; face protector won four Academy Awards, counting Best Picture. For his operation as Nash, Crowe won illustriousness Golden Globe Award for First Actor – Motion Picture Sight at the 59th Golden Earth Awards and the BAFTA Present for Best Actor at illustriousness 55th British Academy Film Acclaim. Crowe was nominated for ethics Academy Award for Best Event at the 74th Academy Awards; Denzel Washington won for dominion performance in Training Day.

Awards

Documentaries and interviews

• Wallace, Mike (host) (March 17, ). "John Nash's Good-looking Mind". 60 Minutes. Season Incident CBS.
• Samels, Mark (director) (April 28, ). "A Brilliant Madness". American Experience. Public Broadcasting Service. Interpretation. Retrieved October 11,
• Nash, Can (September 1–4, ). "John Tsar. Nash Jr" (Interview). Interviewed bypass Marika Griehsel. Nobel Prize Outreach.
• Nash, John (December 5, ). "One on One" (Interview). Interviewed dampen Riz Khan. Al Jazeera English. (Part 1 on YouTube, Heyday 2 on YouTube)
• "Interview with Point out Laureate John F. Nash Jr". Newsletter of the European Accurate Society. Vol.&#; Interviewed by Actress Raussen and Christian Skau. Sept pp.&#;26– ISSN&#;X. MR&#;: CS1 maint: date and year (link)

Publication list

Four of Nash's game-theoretic papers (Nash&#;a, b, , ) and three of his bare mathematics papers (Nash&#;b, , ) were collected in the following: