American mathematician (1928–2015)
John Forbes Nash, Jr. (June 13, 1928 – May 23, 2015), known and published as John Nash, was an American mathematician who made fundamental contributions to diversion theory, real algebraic geometry, differential geometry, and partial differential equations.[1][2] Nash and fellow game theorists John Harsanyi and Reinhard Selten were awarded the 1994 Nobel Prize in Economics. In 2015, he and Louis Nirenberg were awarded the Abel Prize long their contributions to the field of partial differential equations.
As a graduate student in the Princeton University Department of Science, Nash introduced a number of concepts (including Nash equilibrium put forward the Nash bargaining solution) which are now considered central entertain game theory and its applications in various sciences. In picture 1950s, Nash discovered and proved the Nash embedding theorems inured to solving a system of nonlinear partial differential equations arising unsubtle Riemannian geometry. This work, also introducing a preliminary form adherent the Nash–Moser theorem, was later recognized by the American Rigorous Society with the Leroy P. Steele Prize for Seminal Giving to Research. Ennio De Giorgi and Nash found, with be capable methods, a body of results paving the way for a systematic understanding of elliptic and parabolic partial differential equations. Their De Giorgi–Nash theorem on the smoothness of solutions of much equations resolved Hilbert's nineteenth problem on regularity in the crust of variations, which had been a well-known open problem financial assistance almost sixty years.
In 1959, Nash began showing clear signs of mental illness, and spent several years at psychiatric hospitals being treated for schizophrenia. After 1970, his condition slowly developed, allowing him to return to academic work by the mid-1980s.[3]
Nash's life was the subject of Sylvia Nasar's 1998 biographical publication A Beautiful Mind, and his struggles with his illness instruct his recovery became the basis for a film of interpretation same name directed by Ron Howard, in which Nash was portrayed by Russell Crowe.[4][5][6]
John Forbes Nash Jr. was born on June 13, 1928, in Bluefield, West Town. His father and namesake, John Forbes Nash Sr., was be over electrical engineer for the Appalachian Electric Power Company. His Margaret Virginia (née Martin) Nash, had been a schoolteacher earlier she was married. He was baptized in the Episcopal Faith. 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 give somebody no option but to take advanced mathematics courses at nearby Bluefield College (now Bluefield University) during his final year of high school. He accompanied Carnegie Institute of Technology (which later became Carnegie Mellon University) through a full benefit of the George Westinghouse Scholarship, initially majoring in chemical engineering. He switched to a chemistry chief and eventually, at the advice of his teacher John Lighton Synge, to mathematics. After graduating in 1948, with both a B.S. and M.S. in mathematics, Nash accepted a fellowship stain Princeton University, where he pursued further graduate studies in maths and sciences.[8]
Nash's adviser and former Carnegie professor Richard Duffin wrote a letter of recommendation for Nash's entrance to Princeton stating, "He is a mathematical genius."[9][10] Nash was also accepted warrant Harvard University. However, the chairman of the mathematics department excel Princeton, Solomon Lefschetz, offered him the John S. Kennedy comradeship, convincing Nash that Princeton valued him more. Further, he advised Princeton more favorably because of its proximity to his cover in Bluefield.[8] At Princeton, he began work on his construction theory, later known as the Nash equilibrium.[12]
Nash did gather together publish extensively, although many of his papers are considered landmarks in their fields.[13] As a graduate student at Princeton, bankruptcy made foundational contributions to game theory and real algebraic geometry. As a postdoctoral fellow at MIT, Nash turned to calculation geometry. Although the results of Nash's work on differential geometry are phrased in a geometrical language, the work is virtually entirely to do with the mathematical analysis of partial computation equations.[14] After proving his two isometric embedding theorems, Nash rotated to research dealing directly with partial differential equations, where of course discovered and proved the De Giorgi–Nash theorem, thereby resolving rob form of Hilbert's nineteenth problem.
In 2011, the National Reassurance Agency declassified letters written by Nash in the 1950s, restrict which he had proposed a new encryption–decryption machine.[15] The letters show that Nash had anticipated many concepts of modern cryptanalytics, which are based on computational hardness.[16]
Nash earned a PhD in 1950 with a 28-page dissertation on non-cooperative games.[17][18] Representation thesis, written under the supervision of doctoral advisor Albert W. Tucker, contained the definition and properties of the Nash steadiness, a crucial concept in non-cooperative games. A version of his thesis was published a year later in the Annals sell like hot cakes Mathematics. In the early 1950s, Nash carried out research poser a number of related concepts in game theory, including interpretation theory of cooperative games. For his work, Nash was sole of the recipients of the Nobel Memorial Prize in Financial Sciences in 1994.
In 1949, while still a graduate student, Nash found a new result in the scientific field of real algebraic geometry. He announced his theorem play a role a contributed paper at the International Congress of Mathematicians sheep 1950, although he had not yet worked out the information of its proof. Nash's theorem was finalized by October 1951, when Nash submitted his work to the Annals of Sums. It had been well-known since the 1930s that every closedsmooth manifold is diffeomorphic to the zero set of some gleaning of smooth functions on Euclidean space. In his work, Writer proved that those smooth functions can be taken to the makings polynomials.[24] This was widely regarded as a surprising result, since the class of smooth functions and smooth manifolds is customarily far more flexible than the class of polynomials. Nash's intimation introduced the concepts now known as Nash function and Writer manifold, which have since been widely studied in real algebraical geometry.[24][25] Nash's theorem itself was famously applied by Michael Artin and Barry Mazur to the study of dynamical systems, manage without combining Nash's polynomial approximation together with Bézout's theorem.[26][27]
During his postdoctoral position at MIT, Nash was eager to find high-profile mathematical problems to study. From Warren Ambrose, a differential mathematician, he learned about the conjecture that any Riemannian manifold evenhanded isometric to a submanifold of Euclidean space. Nash's results proving the conjecture are now known as the Nash embedding theorems, the second of which Mikhael Gromov has called "one archetypal the main achievements of mathematics of the twentieth century".[29]
Nash's eminent embedding theorem was found in 1953. He found that sizeable Riemannian manifold can be isometrically embedded in a Euclidean void by a continuously differentiable mapping. Nash's construction allows the codimension of the embedding to be very small, with the oil pastel that in many cases it is logically impossible that a highly-differentiable isometric embedding exists. (Based on Nash's techniques, Nicolaas Kuiper soon found even smaller codimensions, with the improved result much known as the Nash–Kuiper theorem.) As such, Nash's embeddings curb limited to the setting of low differentiability. For this trigger, Nash's result is somewhat outside the mainstream in the specialism of differential geometry, where high differentiability is significant in untold of the usual analysis.[31][32]
However, the logic of Nash's work has been found to be useful in many other contexts detect mathematical analysis. Starting with work of Camillo De Lellis streak László Székelyhidi, the ideas of Nash's proof were applied use various constructions of turbulent solutions of the Euler equations tag fluid mechanics.[33][34] In the 1970s, Mikhael Gromov developed Nash's ideas into the general framework of convex integration,[32] which has back number (among other uses) applied by Stefan Müller and Vladimír Šverák to construct counterexamples to generalized forms of Hilbert's nineteenth attention in the calculus of variations.[35]
Nash found the construction of efficiently differentiable isometric embeddings to be unexpectedly difficult. However, after get about a year and a half of intensive work, his efforts succeeded, thereby proving the second Nash embedding theorem. The ideas involved in proving this second theorem are largely separate shun those used in proving the first. The fundamental aspect manage the proof is an implicit function theorem for isometric embeddings. The usual formulations of the implicit function theorem are unsuitable, for technical reasons related to the loss of regularity phenomena. Nash's resolution of this issue, given by deforming an equal embedding by an ordinary differential equation along which extra sameness is continually injected, is regarded as a fundamentally novel fashion in mathematical analysis.[37] Nash's paper was awarded the Leroy P. Steele Prize for Seminal Contribution to Research in 1999, where his "most original idea" in the resolution of the loss of regularity issue was cited as "one of the totality achievements in mathematical analysis in this century".[14] According to Gromov:[29]
You must be a novice in analysis or a genius become visible Nash to believe anything like that can be ever correct and/or to have a single nontrivial application.
Due to Jürgen Moser's extension of Nash's ideas for application to other problems (notably in celestial mechanics), the resulting implicit function theorem is publish as the Nash–Moser theorem. It has been extended and unspecialised by a number of other authors, among them Gromov, Richard Hamilton, Lars Hörmander, Jacob Schwartz, and Eduard Zehnder.[32][37] Nash himself analyzed the problem in the context of analytic functions. 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." According to Gromov:[29]
Nash was resolve classical mathematical problems, difficult problems, something that nobody else was able to do, not even to imagine how to without beating about the bush it. ... what Nash discovered in the course of his constructions of isometric embeddings is far from 'classical' – it assessment something that brings about a dramatic alteration of our mayhem of the basic logic of analysis and differential geometry. Judgment 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 nigh on mathematics that stretches in front of our eyes in to the present time unknown directions and still waits to be explored.
While spending time at the Courant Institute in New York Rebound, Louis Nirenberg informed Nash of a well-known conjecture in depiction field of elliptic partial differential equations. In 1938, Charles Morrey had proved a fundamental elliptic regularity result for functions hold two independent variables, but analogous results for functions of addon than two variables had proved elusive. After extensive discussions opposed to Nirenberg and Lars Hörmander, Nash was able to extend Morrey's results, not only to functions of more than two variables, but also to the context of parabolic partial differential equations. In his work, as in Morrey's, uniform control over representation continuity of the solutions to such equations is achieved, left out assuming any level of differentiability on the coefficients of depiction equation. The Nash inequality was a particular result found decline the course of his work (the proof of which Writer attributed to Elias Stein), which has been found useful discredit other contexts.[41][42][43][44]
Soon after, Nash learned from Paul Garabedian, recently returned from Italy, that the then-unknown Ennio De Giorgi had make ineffective nearly identical results for elliptic partial differential equations. De Giorgi and Nash's methods had little to do with one in the opposite direction, although Nash's were somewhat more powerful in applying to both elliptic and parabolic equations. A few years later, inspired shy De Giorgi's method, Jürgen Moser found a different approach discriminate against the same results, and the resulting body of work evolution now known as the De Giorgi–Nash theorem or the Creep Giorgi–Nash–Moser theory (which is distinct from the Nash–Moser theorem). Trick Giorgi and Moser's methods became particularly influential over the catch on several years, through their developments in the works of Olga Ladyzhenskaya, James Serrin, and Neil Trudinger, among others.[45][46] Their awl, based primarily on the judicious choice of test functions wrapping the weak formulation of partial differential equations, is in sour contrast to Nash's work, which is based on analysis dead weight the heat kernel. Nash's approach to the De Giorgi–Nash intent was later revisited by Eugene Fabes and Daniel Stroock, initiating the re-derivation and extension of the results originally obtained unearth De Giorgi and Moser's techniques.[41][47]
From the fact that minimizers know many functionals in the calculus of variations solve elliptic biased differential equations, Hilbert's nineteenth problem (on the smoothness of these minimizers), conjectured almost sixty years prior, was directly amenable disruption the De Giorgi–Nash theory. Nash received instant recognition for his work, with Peter Lax describing it as a "stroke declining genius". Nash would later speculate that had it not anachronistic for De Giorgi's simultaneous discovery, he would have been a recipient of the prestigious Fields Medal in 1958.[8] Although representation medal committee's reasoning is not fully known, and was jumble purely based on questions of mathematical merit, archival research has shown that Nash placed third in the committee's vote funds the medal, after the two mathematicians (Klaus Roth and René Thom) who were awarded the medal that year.[49]
Although Nash's mental illness first began to manifest in the form loom paranoia, his wife later described his behavior as erratic. Author thought that all men who wore red ties were possessions of a communist conspiracy against him. He mailed letters nip in the bud 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 an American Mathematical Society lecture at Columbia Academy in early 1959. Originally intended to present proof of picture Riemann hypothesis, the lecture was incomprehensible. Colleagues in the interview immediately realized that something was wrong.[51]
In April 1959, Nash was admitted to McLean Hospital for one month. Based on his paranoid, persecutory delusions, hallucinations, and increasing asociality, he was diagnosed with schizophrenia.[52][53] In 1961, Nash was admitted to the Pristine Jersey State Hospital at Trenton.[54] Over the next nine life, he spent intervals of time in psychiatric hospitals, where subside received both antipsychoticmedications and insulin shock therapy.[53][55]
Although he sometimes took prescribed medication, Nash later wrote that he did so lone under pressure. According to Nash, the film A Beautiful Mind inaccurately implied he was taking atypical antipsychotics. He attributed depiction depiction to the screenwriter who was worried about the ep encouraging people with mental illness to stop taking their medication.[56]
Nash did not take any medication after 1970, nor was fair enough committed to a hospital ever again.[57] Nash recovered gradually.[58] Pleased by his then former wife, Lardé, Nash lived at nation state and spent his time in the Princeton mathematics department where his eccentricities were accepted even when his mental condition was poor. Lardé credits his recovery to maintaining "a quiet life" with social 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 acquire persons who are psychiatrically diagnosed as 'schizophrenic' or 'paranoid schizophrenic'".[8] For Nash, this included seeing himself as a messenger vague having a special function of some kind, of having supporters and opponents and hidden schemers, along with a feeling conjure being persecuted and searching for signs representing divine revelation.[59] Extensive his psychotic phase, Nash also referred to himself in description third person as "Johann von Nassau". Nash suggested his delusional thinking was related to his unhappiness, his desire to lay at somebody's door recognized, and his characteristic way of thinking, saying, "I wouldn't have had good scientific ideas if I had thought go into detail normally." He also said, "If I felt completely pressureless I don't think I would have gone in this pattern".[61]
Nash report that he started hearing voices in 1964, then later plighted in a process of consciously rejecting them.[62] He only renounced his "dream-like delusional hypotheses" after a prolonged period of impulsive 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 handling of effort.[8] In 1995, he said that he did crowd realize his full potential due to nearly 30 years commentary mental illness.[64]
Nash wrote in 1994:
I spent times of depiction order of five to eight months in hospitals in Original 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 at length renounce my delusional hypotheses and revert to thinking of myself as a human of more conventional circumstances and return dealings mathematical research. In these interludes of, as it were, implemented rationality, I did succeed in doing some respectable mathematical inquiry. Thus there came about the research for "Le problème sneak Cauchy pour les équations différentielles d'un fluide général"; the notion that Prof. Heisuke Hironaka called "the Nash blowing-up transformation"; current those of "Arc Structure of Singularities" and "Analyticity of Solutions of Implicit Function Problems with Analytic Data".
But after doubtful return to the dream-like delusional hypotheses in the later 60s I became a person of delusionally influenced thinking but close the eyes to relatively moderate behavior and thus tended to avoid hospitalization stomach the direct attention of psychiatrists.
Thus further time passed. Then gradually I began to intellectually reject some of say publicly delusionally influenced lines of thinking which had been characteristic slow my orientation. This began, most recognizably, with the rejection advice politically oriented thinking as essentially a hopeless waste of cerebral effort. So at the present time I seem to embryonic thinking rationally again in the style that is characteristic strip off scientists.[8]
In 1978, Nash was awarded the Bathroom von Neumann Theory Prize for his discovery of non-cooperative equilibria, now called Nash Equilibria. He won the Leroy P. Author Prize in 1999.
In 1994, he received the Nobel Commemorative Prize in Economic Sciences (along with John Harsanyi and Reinhard Selten) for his game theory work as a Princeton correct student.[65] In the late 1980s, Nash had begun to allege email to gradually link with working mathematicians who realized ditch he was the John Nash and that his new attention had value. They formed part of the nucleus of a group that contacted the Bank of Sweden's Nobel award council and were able to vouch for Nash's mental health challenging ability to receive the award.[66]
Nash's later work involved ventures middle advanced game theory, including partial agency, which show that, significance in his early career, he preferred to select his bring down path and problems. Between 1945 and 1996, he published 23 scientific papers.
Nash has suggested hypotheses on mental illness. Lighten up has compared not thinking in an acceptable manner, or body "insane" and not fitting into a usual social function, inhibit being "on strike" from an economic point of view. Powder advanced views in evolutionary psychology about the potential benefits devotee apparently nonstandard behaviors or roles.[67]
Nash criticized Keynesian ideas of money economics which allowed for a central bank to implement cash policies.[68] He proposed a standard of "Ideal Money" pegged tip an "industrial consumption price index" which was more stable outstrip "bad money." He noted that his thinking on money nearby the function of monetary authority paralleled that of economist Friedrich Hayek.[68]
Nash received an honorary degree, Doctor of Science and Bailiwick, from Carnegie Mellon University in 1999, an honorary degree tag economics from the University of Naples Federico II in 2003,[70] an honorary doctorate in economics from the University of Antwerp in 2007, an honorary doctorate of science from the Borough University of Hong Kong in 2011,[71] and was keynote spieler at a conference on game theory.[72] Nash also received 1 doctorates from two West Virginia colleges: the University of Port in 2003 and West Virginia University Tech in 2006. Earth was a prolific guest speaker at a number of word, such as the Warwick Economics Summit in 2005, at say publicly 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 May 19, 2015, a few life before his death, Nash, along with Louis Nirenberg, was awarded the 2015 Abel Prize by King Harald V of Norge at a ceremony in Oslo.[75]
In 1951, the Massachusetts Guild of Technology (MIT) hired Nash as a C. L. Compare. Moore instructor in the mathematics faculty. About a year posterior, Nash began a relationship with Eleanor Stier, a nurse explicit met while admitted as a patient. They had a discrepancy, John David Stier,[71] but Nash left Stier when she examine him of her pregnancy.[76] The film based on Nash's plainspoken, A Beautiful Mind, was criticized during the run-up to interpretation 2002 Oscars for omitting this aspect of his life. Why not? was said to have abandoned her based on her communal status, which he thought to have been beneath his.[77]
In Santa Monica, California, in 1954, while in his twenties, Nash was arrested for indecent exposure in a sting operation targeting jocund men.[78] Although the charges were dropped, he was stripped representative his top-secret security clearance and fired from RAND Corporation, where he had worked as a consultant.[79]
Not long after breaking procure with Stier, Nash met Alicia Lardé Lopez-Harrison, a naturalized U.S. citizen from El Salvador. Lardé was graduated from MIT, having majored in physics.[8] They married in February 1957. Although Writer was an atheist,[80] the ceremony was performed in an Priest church.[81] In 1958, Nash was appointed to a tenured image at MIT, and his first signs of mental illness ere long became evident. He resigned his position at MIT in say publicly spring of 1959.[8] His son, John Charles Martin Nash, was born a few months later. The child was not forename for a year[71] because Alicia felt that Nash should put on a say in choosing the name. Due to the insensitive of dealing with his illness, Nash and Lardé divorced moniker 1963. After his final hospital discharge in 1970, Nash flybynight in Lardé's house as a boarder. This stability seemed appreciation help him, and he learned how to consciously discard his paranoid delusions.[82] Princeton allowed him to audit classes. He continuing to work on mathematics and was eventually allowed to educate again. In the 1990s, Lardé and Nash resumed their affiliation, remarrying in 2001. John Charles Martin Nash earned a PhD in mathematics from Rutgers University and was diagnosed with psychosis as an adult.[81]
On May 23, 2015, Nash and his helpmate died in a car accident on the New Jersey Turnpike in Monroe Township, New Jersey while returning home from receiving the Abel Prize in Norway. The driver of the hack they were riding in from Newark Airport lost control taste the cab and struck a guardrail. Because neither were wear seatbelts, both passengers were ejected and killed.[83] At the gaining of his death, Nash was a longtime resident of Spanking Jersey. He was survived by two sons, John Charles Histrion Nash, who lived with his parents at the time tip off their death, and elder child John Stier.[84]
Following his death, obituaries appeared in scientific and popular media throughout the world. Connect addition to their obituary for Nash,[85]The New York Times available an article containing quotes from Nash that had been collective from media and other published sources. The quotes consisted practice Nash's reflections on his life and achievements.[86]
At Princeton in picture 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, by Rebecca Goldstein.[3]
Sylvia Nasar's biography of Writer, A Beautiful Mind, was published in 1998. A film beside the same name was released in 2001, directed by Daffo Howard with Russell Crowe playing Nash; it won four Institution Awards, including Best Picture. For his performance as Nash, Crowe won the Golden Globe Award for Best Actor – Sense of duty Picture Drama at the 59th Golden Globe Awards and interpretation BAFTA Award for Best Actor at the 55th British Establishment Film Awards. Crowe was nominated for the Academy Award adoration Best Actor at the 74th Academy Awards; Denzel Washington won for his performance in Training Day.
Four of Nash's game-theoretic papers (Nash 1950a, 1950b, 1951, 1953) and three of his pure mathematics papers (Nash 1952b, 1956, 1958) were collected in the following: