John Forbes Nash Jr. (June 13, 1928 – May 23, 2015) was an American mathematician who made fundamental contributions to game theory,

real algebraic geometry In mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them (in particular real polynomia ...

, differential geometry, and partial differential equations. Nash and fellow game theorists

John Harsanyi John Charles Harsanyi ( hu, Harsányi János Károly; May 29, 1920 – August 9, 2000) was a Hungarian-American economist and the recipient of the Nobel Memorial Prize in Economic Sciences in 1994. He is best known for his contributions to the ...

and

Reinhard Selten Reinhard Justus Reginald Selten (; 5 October 1930 – 23 August 2016) was a German economist, who won the 1994 Nobel Memorial Prize in Economic Sciences (shared with John Harsanyi and John Nash). He is also well known for his work in boun ...

were awarded the 1994

Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel The Nobel Memorial Prize in Economic Sciences, officially the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel ( sv, Sveriges riksbanks pris i ekonomisk vetenskap till Alfred Nobels minne), is an economics award administered ...

(popularly known as the Nobel Prize in Economics). In 2015, he and

Louis Nirenberg Louis Nirenberg (February 28, 1925 – January 26, 2020) was a Canadian-American mathematician, considered one of the most outstanding mathematicians of the 20th century. Nearly all of his work was in the field of partial differential equat ...

were awarded the Abel Prize for their contributions to the field of partial differential equations. As a graduate student in the Mathematics Department at

Princeton University Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ...

, Nash introduced a number of concepts (including Nash equilibrium and the

Nash bargaining solution Cooperative bargaining is a process in which two people decide how to share a surplus that they can jointly generate. In many cases, the surplus created by the two players can be shared in many ways, forcing the players to negotiate which division o ...

) which are now considered central to game theory and its applications in various sciences. In the 1950s, Nash discovered and proved the Nash embedding theorems by solving a system of nonlinear partial differential equations arising in

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...

. This work, also introducing a preliminary form of the

Nash–Moser theorem In the mathematical field of analysis, the Nash–Moser theorem, discovered by mathematician John Forbes Nash and named for him and Jürgen Moser, is a generalization of the inverse function theorem on Banach spaces to settings when the required so ...

, was later recognized by the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

with the Leroy P. Steele Prize for Seminal Contribution to Research. Ennio De Giorgi and Nash found, with separate methods, a body of results paving the way for a systematic understanding of

elliptic In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...

and parabolic partial differential equations. Their De Giorgi–Nash theorem on the smoothness of solutions of such equations resolved Hilbert's nineteenth problem on regularity in the calculus of variations, which had been a well-known open problem for almost sixty years. In 1959, Nash began showing clear signs of mental illness, and spent several years at

psychiatric hospital Psychiatric hospitals, also known as mental health hospitals, behavioral health hospitals, are hospitals or wards specializing in the treatment of severe mental disorders, such as schizophrenia, bipolar disorder, eating disorders, dissociat ...

s being treated for

schizophrenia Schizophrenia is a mental disorder characterized by continuous or relapsing episodes of psychosis. Major symptoms include hallucinations (typically hearing voices), delusions, and disorganized thinking. Other symptoms include social wit ...

. After 1970, his condition slowly improved, allowing him to return to academic work by the mid-1980s. His struggles with his illness and his recovery became the basis for Sylvia Nasar's biographical book '' A Beautiful Mind'' in 1998, as well as a film of the same name directed by

Ron Howard Ronald William Howard (born March 1, 1954) is an American director, producer, screenwriter, and actor. He first came to prominence as a child actor, guest-starring in several television series, including an episode of ''The Twilight Zone''. He ...

, in which Nash was portrayed by New Zealand Australian actor Russell Crowe.

Early life and education

John Forbes Nash Jr. was born on June 13, 1928, in

Bluefield, West Virginia Bluefield is a city in Mercer County, West Virginia, United States. The population was 9,658 at the 2020 census. It is the principal city of the Bluefield WV- VA micropolitan area, which had a population of 106,363 in 2020. Geography Bluefie ...

. His father and namesake, John Forbes Nash Sr., was an electrical engineer for the Appalachian Electric Power Company. His mother, Margaret Virginia (née Martin) Nash, had been a schoolteacher before she was married. He was baptized in the Episcopal Church. He had a younger sister, Martha (born November 16, 1930). Nash attended kindergarten and public school, and he learned from books provided by his parents and grandparents. Nash's parents pursued opportunities to supplement their son's education, and arranged for him to take advanced mathematics courses at a local community college during his final year of high school. He attended

Carnegie Institute of Technology Carnegie Mellon University (CMU) is a private research university in Pittsburgh, Pennsylvania. One of its predecessors was established in 1900 by Andrew Carnegie as the Carnegie Technical Schools; it became the Carnegie Institute of Technology ...

(which later became Carnegie Mellon University) through a full benefit of the George Westinghouse Scholarship, initially majoring in

chemical engineering Chemical engineering is an engineering field which deals with the study of operation and design of chemical plants as well as methods of improving production. Chemical engineers develop economical commercial processes to convert raw materials int ...

. He switched to a chemistry major and eventually, at the advice of his teacher

John Lighton Synge John Lighton Synge (; 23 March 1897 – 30 March 1995) was an Irish mathematician and physicist, whose seven-decade career included significant periods in Ireland, Canada, and the USA. He was a prolific author and influential mentor, and is cre ...

, to mathematics. After graduating in 1948, with both a

B.S. A Bachelor of Science (BS, BSc, SB, or ScB; from the Latin ') is a bachelor's degree awarded for programs that generally last three to five years. The first university to admit a student to the degree of Bachelor of Science was the University ...

and

M.S. A Master of Science ( la, Magisterii Scientiae; abbreviated MS, M.S., MSc, M.Sc., SM, S.M., ScM or Sc.M.) is a master's degree in the field of science awarded by universities in many countries or a person holding such a degree. In contrast to ...

in mathematics, Nash accepted a fellowship to

, where he pursued further

graduate studies Postgraduate or graduate education refers to academic or professional degrees, certificates, diplomas, or other qualifications pursued by post-secondary students who have earned an undergraduate (bachelor's) degree. The organization and stru ...

in mathematics and sciences. Nash's adviser and former Carnegie professor

Richard Duffin Richard James Duffin (1909 – October 29, 1996) was an American physicist, known for his contributions to electrical transmission theory and to the development of geometric programming and other areas within operations research. Education and ...

wrote a letter of recommendation for Nash's entrance to Princeton stating, "He is a mathematical genius". Nash was also accepted at

Harvard University Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of high ...

. However, the chairman of the mathematics department at Princeton,

Solomon Lefschetz Solomon Lefschetz (russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear o ...

, offered him the John S. Kennedy fellowship, convincing Nash that Princeton valued him more. Further, he considered Princeton more favorably because of its proximity to his family in Bluefield. At Princeton, he began work on his equilibrium theory, later known as the Nash equilibrium.

Research contributions

Nash did not publish extensively, although many of his papers are considered landmarks in their fields. As a graduate student at Princeton, he made foundational contributions to game theory and

. As a postdoctoral fellow at

MIT The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the m ...

, Nash turned to differential geometry. Although the results of Nash's work on differential geometry are phrased in a geometrical language, the work is almost entirely to do with the

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...

partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...

. After proving his two isometric embedding theorems, Nash turned to research dealing directly with partial differential equations, where he discovered and proved the De Giorgi–Nash theorem, thereby resolving one form of Hilbert's nineteenth problem. In 2011, the

National Security Agency The National Security Agency (NSA) is a national-level intelligence agency of the United States Department of Defense, under the authority of the Director of National Intelligence (DNI). The NSA is responsible for global monitoring, collect ...

declassified letters written by Nash in the 1950s, in which he had proposed a new

encryption In cryptography, encryption is the process of encoding information. This process converts the original representation of the information, known as plaintext, into an alternative form known as ciphertext. Ideally, only authorized parties can de ...

–decryption machine. The letters show that Nash had anticipated many concepts of modern

cryptography Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adver ...

, which are based on computational hardness.

Game theory

Nash earned a PhD in 1950 with a 28-page dissertation on non-cooperative games. The thesis, written under the supervision of doctoral advisor Albert W. Tucker, contained the definition and properties of the Nash equilibrium, a crucial concept in non-cooperative games. A version of his thesis was published a year later in the

Annals of Mathematics The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the ...

. In the early 1950s, Nash carried out research on a number of related concepts in game theory, including the theory of cooperative games. For his work, Nash was one of the recipients of the

Nobel Memorial Prize in Economic Sciences The Nobel Memorial Prize in Economic Sciences, officially the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel ( sv, Sveriges riksbanks pris i ekonomisk vetenskap till Alfred Nobels minne), is an economics award administered ...

in 1994.

Real algebraic geometry

In 1949, while still a graduate student, Nash found a new result in the mathematical field of

. He announced his theorem in a contributed paper at the International Congress of Mathematicians in 1950, although he had not yet worked out the details of its proof. Nash's theorem was finalized by October 1951, when Nash submitted his work to the

. It had been well-known since the 1930s that every closed

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...

diffeomorphic In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two man ...

to the

zero set In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or e ...

of some collection of smooth functions on

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...

. In his work, Nash proved that those smooth functions can be taken to be

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...

s. This was widely regarded as a surprising result, since the class of smooth functions and smooth manifolds is usually far more flexible than the class of polynomials. Nash's proof introduced the concepts now known as Nash function and Nash manifold, which have since been widely studied in real algebraic geometry. Nash's theorem itself was famously applied by

Michael Artin Michael Artin (; born 28 June 1934) is a German-American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry.Barry Mazur to the study of

dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...

s, by combining Nash's polynomial approximation together with

Bézout's theorem Bézout's theorem is a statement in algebraic geometry concerning the number of common zeros of polynomials in indeterminates. In its original form the theorem states that ''in general'' the number of common zeros equals the product of the deg ...

Differential geometry

During his postdoctoral position at

, Nash was eager to find high-profile mathematical problems to study. From Warren Ambrose, a

differential geometer Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilin ...

, he learned about the conjecture that any Riemannian manifold is isometric to a

submanifold In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...

. Nash's results proving the conjecture are now known as the

Nash embedding theorem The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash Jr., state that every Riemannian manifold can be isometrically embedding, embedded into some Euclidean space. Isometry, Isometric means preserving the length of every ...

s, the second of which Mikhael Gromov has called "one of the main achievements of mathematics of the twentieth century". Nash's first embedding theorem was found in 1953. He found that any Riemannian manifold can be isometrically embedded in a Euclidean space by a continuously differentiable mapping. Nash's construction allows the codimension of the embedding to be very small, with the effect that in many cases it is logically impossible that a highly-differentiable isometric embedding exists. (Based on Nash's techniques,

Nicolaas Kuiper Nicolaas Hendrik Kuiper (; 28 June 1920 – 12 December 1994) was a Dutch mathematician, known for Kuiper's test and proving Kuiper's theorem. He also contributed to the Nash embedding theorem. Kuiper studied at University of Leiden in 1937-4 ...

soon found even smaller codimensions, with the improved result often known as the ''Nash–Kuiper theorem''.) As such, Nash's embeddings are limited to the setting of low differentiability. For this reason, Nash's result is somewhat outside the mainstream in the field of differential geometry, where high differentiability is significant in much of the usual analysis. However, the logic of Nash's work has been found to be useful in many other contexts in

. Starting with work of

Camillo De Lellis Camillo De Lellis (born 11 June 1976) is an Italian mathematician who is active in the fields of calculus of variations, hyperbolic systems of conservation laws, geometric measure theory and fluid dynamics. He is a permanent faculty member i ...

and László Székelyhidi, the ideas of Nash's proof were applied for various constructions of turbulent solutions of the

Euler equations 200px, Leonhard Euler (1707–1783) In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include ...

fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...

. In the 1970s, Mikhael Gromov developed Nash's ideas into the general framework of ''convex integration'', which has been (among other uses) applied by Stefan Müller and

Vladimír Šverák Vladimír Šverák (born 1959) is a Czech mathematician. Since 1990, he has been a professor at the University of Minnesota. Šverák made notable contributions to calculus of variations. Šverák obtained his doctorate from the Charles Universit ...

to construct counterexamples to generalized forms of Hilbert's nineteenth problem in the calculus of variations. Nash found the construction of smoothly differentiable isometric embeddings to be unexpectedly difficult. However, after around 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 from those used in proving the first. The fundamental aspect of the proof is an implicit function theorem for isometric embeddings. The usual formulations of the implicit function theorem are inapplicable, for technical reasons related to the ''loss of regularity'' phenomena. Nash's resolution of this issue, given by deforming an isometric embedding by an

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...

along which extra regularity is continually injected, is regarded as a fundamentally novel technique in

. 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 great achievements in mathematical analysis in this century". According to Gromov: Due to

Jürgen Moser Jürgen Kurt Moser (July 4, 1928 – December 17, 1999) was a German-American mathematician, honored for work spanning over four decades, including Hamiltonian dynamical systems and partial differential equations. Life Moser's mother Ilse Strehl ...

's extension of Nash's ideas for application to other problems (notably in

celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...

), the resulting implicit function theorem is known as the

. It has been extended and generalized by a number of other authors, among them Gromov, Richard Hamilton,

Lars Hörmander Lars Valter Hörmander (24 January 1931 – 25 November 2012) was a Swedish mathematician who has been called "the foremost contributor to the modern theory of linear partial differential equations". Hörmander was awarded the Fields Medal ...

, Jacob Schwartz, and Eduard Zehnder. Nash himself analyzed the problem in the context of

analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...

s. 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:

Partial differential equations

While spending time at the

Courant Institute The Courant Institute of Mathematical Sciences (commonly known as Courant or CIMS) is the mathematics research school of New York University (NYU), and is among the most prestigious mathematics schools and mathematical sciences research cente ...

in New York City,

informed Nash of a well-known conjecture in the field of elliptic partial differential equations. In 1938, Charles Morrey had proved a fundamental

elliptic regularity In the theory of partial differential equations, a partial differential operator P defined on an open subset :U \subset^n is called hypoelliptic if for every distribution u defined on an open subset V \subset U such that Pu is C^\infty (smooth ...

result for functions of two independent variables, but analogous results for functions of more than two variables had proved elusive. After extensive discussions with Nirenberg and

, 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 the continuity of the solutions to such equations is achieved, without assuming any level of differentiability on the coefficients of the equation. The Nash inequality was a particular result found in the course of his work (the proof of which Nash attributed to

Elias Stein Elias Menachem Stein (January 13, 1931 – December 23, 2018) was an American mathematician who was a leading figure in the field of harmonic analysis. He was the Albert Baldwin Dod Professor of Mathematics, Emeritus, at Princeton University, whe ...

), which has been found useful in other contexts. Soon after, Nash learned from Paul Garabedian, recently returned from Italy, that the then-unknown Ennio De Giorgi had found nearly identical results for elliptic partial differential equations. De Giorgi and Nash's methods had little to do with one another, although Nash's were somewhat more powerful in applying to both elliptic and parabolic equations. A few years later, inspired by De Giorgi's method,

found a different approach to the same results, and the resulting body of work is now known as the De Giorgi–Nash theorem or the De Giorgi–Nash–Moser theory (which is distinct from the

). De Giorgi and Moser's methods became particularly influential over the next several years, through their developments in the works of

Olga Ladyzhenskaya Olga Aleksandrovna Ladyzhenskaya (russian: Óльга Алекса́ндровна Лады́женская, link=no, p=ˈolʲɡə ɐlʲɪˈksandrəvnə ɫɐˈdɨʐɨnskəɪ̯ə, a=Ru-Olga Aleksandrovna Ladyzhenskaya.wav; 7 March 1922 – 12 Jan ...

James Serrin James Burton Serrin (1 November 1926, Chicago, Illinois – 23 August 2012, Minneapolis, Minnesota) was an American mathematician, and a professor at University of Minnesota. Life He received his doctorate from Indiana University in 1951 under t ...

, and

Neil Trudinger Neil Sidney Trudinger (born 20 June 1942) is an Australian mathematician, known particularly for his work in the field of nonlinear elliptic partial differential equations. After completing his B.Sc at the University of New England (Australia) ...

, among others. Their work, based primarily on the judicious choice of

test function Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives d ...

s in the

weak formulation Weak formulations are important tools for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other fields such as partial differential equations. In a weak formulation, equations or con ...

of partial differential equations, is in strong contrast to Nash's work, which is based on analysis of the heat kernel. Nash's approach to the De Giorgi–Nash theory was later revisited by Eugene Fabes and Daniel Stroock, initiating the re-derivation and extension of the results originally obtained from De Giorgi and Moser's techniques. From the fact that minimizers to many functionals in the calculus of variations solve elliptic partial differential equations, Hilbert's nineteenth problem (on the smoothness of these minimizers), conjectured almost sixty years prior, was directly amenable to the De Giorgi–Nash theory. Nash received instant recognition for his work, with

Peter Lax Peter David Lax (born Lax Péter Dávid; 1 May 1926) is a Hungarian-born American mathematician and Abel Prize laureate working in the areas of pure and applied mathematics. Lax has made important contributions to integrable systems, fluid ...

describing it as a "stroke of genius". Nash would later speculate that had it not been for De Giorgi's simultaneous discovery, he would have been a recipient of the prestigious Fields Medal in 1958. Although the medal committee's reasoning is not fully known, and was not purely based on questions of mathematical merit, archival research has shown that Nash placed third in the committee's vote for the medal, after the two mathematicians (

Klaus Roth Klaus Friedrich Roth (29 October 1925 – 10 November 2015) was a German-born British mathematician who won the Fields Medal for proving Roth's theorem on the Diophantine approximation of algebraic numbers. He was also a winner of the De Mo ...

and René Thom) who were awarded the medal that year.

Mental illness

Although Nash's mental illness first began to manifest in the form of

paranoia Paranoia is an instinct or thought process that is believed to be heavily influenced by anxiety or fear, often to the point of delusion and irrationality. Paranoid thinking typically includes persecutory beliefs, or beliefs of conspiracy co ...

, his wife later described his behavior as erratic. Nash thought that all men who wore red ties were part of a communist conspiracy against him. He mailed letters to embassies in Washington, D.C., declaring that they were establishing a government. Nash's psychological issues crossed into his professional life when he gave an

lecture at

Columbia University Columbia University (also known as Columbia, and officially as Columbia University in the City of New York) is a private research university in New York City. Established in 1754 as King's College on the grounds of Trinity Church in Manhatt ...

in early 1959. Originally intended to present proof of the Riemann hypothesis, the lecture was incomprehensible. Colleagues in the audience immediately realized that something was wrong. In April 1959, Nash was admitted to

McLean Hospital McLean Hospital () (formerly known as Somerville Asylum and Charlestown Asylum) is a psychiatric hospital in Belmont, Massachusetts. It is noted for its clinical staff expertise and neuroscience research and is also known for the large number of ...

for one month. Based on his paranoid, persecutory delusions, hallucinations, and increasing

asociality Asociality refers to the lack of motivation to engage in social interaction, or a preference for solitary activities. Asociality may be associated with avolition, but it can, moreover, be a manifestation of limited opportunities for social relati ...

, he was diagnosed with

. Nasar (2011), p. 32. In 1961, Nash was admitted to the New Jersey State Hospital at Trenton. Over the next nine years, he spent intervals of time in

s, where he received both

antipsychotic Antipsychotics, also known as neuroleptics, are a class of psychotropic medication primarily used to manage psychosis (including delusions, hallucinations, paranoia or disordered thought), principally in schizophrenia but also in a range of ...

medications and

insulin shock therapy Insulin shock therapy or insulin coma therapy was a form of psychiatric treatment in which patients were repeatedly injected with large doses of insulin in order to produce daily comas over several weeks.Neustatter WL (1948) ''Modern psychiatry ...

. Although he sometimes took prescribed medication, Nash later wrote that he did so only under pressure. According to Nash, the film ''A Beautiful Mind'' inaccurately implied he was taking

atypical antipsychotic The atypical antipsychotics (AAP), also known as second generation antipsychotics (SGAs) and serotonin–dopamine antagonists (SDAs), are a group of antipsychotic drugs (antipsychotic drugs in general are also known as major tranquilizers and ne ...

s. He attributed the depiction to the screenwriter who was worried about the film encouraging people with mental illness to stop taking their medication. Nash did not take any medication after 1970, nor was he committed to a hospital ever again. Nash recovered gradually. Encouraged by his then former wife, de Lardé, Nash lived at home and spent his time in the Princeton mathematics department where his eccentricities were accepted even when his mental condition was poor. De Lardé credits his

recovery Recovery or Recover may refer to: Arts and entertainment Books * ''Recovery'' (novel), a Star Wars e-book * Recovery Version, a translation of the Bible with footnotes published by Living Stream Ministry Film and television * ''Recovery'' (fil ...

to maintaining "a quiet life" with social support. Nash dated the start of what he termed "mental disturbances" to the early months of 1959, when his wife was pregnant. He described a process of change "from scientific rationality of thinking into the delusional thinking characteristic of persons who are psychiatrically diagnosed as 'schizophrenic' or 'paranoid schizophrenic. For Nash, this included seeing himself as a messenger or having a special function of some kind, of having supporters and opponents and hidden schemers, along with a feeling of being persecuted and searching for signs representing divine revelation. Nash suggested his delusional thinking was related to his unhappiness, his desire to be recognized, and his characteristic way of thinking, saying, "I wouldn't have had good scientific ideas if I had thought more normally." He also said, "If I felt completely pressureless I don't think I would have gone in this pattern". Nash reported that he started hearing voices in 1964, then later engaged in a process of consciously rejecting them. He only renounced his "dream-like delusional hypotheses" after a prolonged period of involuntary commitment in mental hospitals—"enforced rationality". Upon doing so, he was temporarily able to return to productive work as a mathematician. By the late 1960s, he relapsed. Eventually, he "intellectually rejected" his " influenced" and "politically oriented" thinking as a waste of effort. In 1995, he said that he didn't realize his full potential due to nearly 30 years of mental illness.Nash, Joh
"John Nash: My experience with mental illness"
PBS Interview, 2002. Nash wrote in 1994:

Recognition and later career

In 1978, Nash was awarded the

John von Neumann Theory Prize The John von Neumann Theory Prize of the Institute for Operations Research and the Management Sciences (INFORMS) is awarded annually to an individual (or sometimes a group) who has made fundamental and sustained contributions to theory in operat ...

for his discovery of non-cooperative equilibria, now called Nash Equilibria. He won the

Leroy P. Steele Prize The Leroy P. Steele Prizes are awarded every year by the American Mathematical Society, for distinguished research work and writing in the field of mathematics. Since 1993, there has been a formal division into three categories. The prizes have b ...

in 1999. In 1994, he received the

(along with

and

) for his game theory work as a Princeton graduate student. In the late 1980s, Nash had begun to use email to gradually link with working mathematicians who realized that he was John Nash and that his new work had value. They formed part of the nucleus of a group that contacted the

Bank of Sweden Sveriges Riksbank, or simply the ''Riksbank'', is the central bank of Sweden. It is the world's oldest central bank and the fourth oldest bank in operation. Etymology The first part of the word ''riksbank'', ''riks'', stems from the Swedish ...

's Nobel award committee and were able to vouch for Nash's mental health and ability to receive the award. Nash's later work involved ventures in advanced game theory, including partial agency, which show that, as in his early career, he preferred to select his own path and problems. Between 1945 and 1996, he published 23 scientific studies. Nash has suggested hypotheses on mental illness. He has compared not thinking in an acceptable manner, or being "insane" and not fitting into a usual social function, to being "on

strike Strike may refer to: People * Strike (surname) Physical confrontation or removal *Strike (attack), attack with an inanimate object or a part of the human body intended to cause harm *Airstrike, military strike by air forces on either a suspected ...

" from an economic point of view. He advanced views in

evolutionary psychology Evolutionary psychology is a theoretical approach in psychology that examines cognition and behavior from a modern evolutionary perspective. It seeks to identify human psychological adaptations with regards to the ancestral problems they evol ...

about the potential benefits of apparently nonstandard behaviors or roles. Nash developed work on the role of money in society. He criticized interest groups that promote quasi-doctrines based on

Keynesian economics Keynesian economics ( ; sometimes Keynesianism, named after British economist John Maynard Keynes) are the various macroeconomic theories and models of how aggregate demand (total spending in the economy) strongly influences economic output a ...

that permit manipulative short-term inflation and

debt Debt is an obligation that requires one party, the debtor, to pay money or other agreed-upon value to another party, the creditor. Debt is a deferred payment, or series of payments, which differentiates it from an immediate purchase. The ...

tactics that ultimately undermine currencies. He suggested a global "industrial consumption price index" system that would support the development of more "ideal money" that people could trust rather than more unstable "bad money." He noted that some of his thinking parallels that of economist and

political philosopher Political philosophy or political theory is the philosophical study of government, addressing questions about the nature, scope, and legitimacy of public agents and institutions and the relationships between them. Its topics include politics, l ...

Friedrich Hayek Friedrich August von Hayek ( , ; 8 May 189923 March 1992), often referred to by his initials F. A. Hayek, was an Austrian–British economist, legal theorist and philosopher who is best known for his defense of classical liberalism. Haye ...

, regarding money and an atypical viewpoint of the function of authority. Nash received an honorary degree, Doctor of Science and Technology, from Carnegie Mellon University in 1999, an honorary degree in economics from the

University of Naples Federico II The University of Naples Federico II ( it, Università degli Studi di Napoli Federico II) is a public university in Naples, Italy. Founded in 1224, it is the oldest public non-sectarian university in the world, and is now organized into 26 depar ...

in 2003, an honorary doctorate in economics from the

University of Antwerp The University of Antwerp ( nl, Universiteit Antwerpen) is a major Belgian university located in the city of Antwerp. The official abbreviation is ''UA'', but ''UAntwerpen'' is more recently used. The University of Antwerp has about 20,000 stud ...

in 2007, an honorary doctorate of science from the

City University of Hong Kong City University of Hong Kong (CityU) is a world-class public research university located in Kowloon Tong, Hong Kong. It was founded in 1984 as City Polytechnic of Hong Kong and became a fully accredited university in 1994. Currently, CityU is ...

in 2011, and was keynote speaker at a conference on game theory. Nash also received honorary doctorates from two West Virginia colleges: the University of Charleston in 2003 and West Virginia University Tech in 2006. He was a prolific guest speaker at a number of events, such as the Warwick Economics Summit in 2005, at the

University of Warwick , mottoeng = Mind moves matter , established = , type = Public research university , endowment = £7.0 million (2021) , budget = £698.2 million (2020� ...

. Nash was elected to the

American Philosophical Society The American Philosophical Society (APS), founded in 1743 in Philadelphia, is a scholarly organization that promotes knowledge in the sciences and humanities through research, professional meetings, publications, library resources, and communit ...

in 2006 and became a fellow of the American Mathematical Society in 2012. On May 19, 2015, a few days before his death, Nash, along with

, was awarded the 2015 Abel Prize by King

Harald V of Norway Harald V ( no, Harald den femte, ; born 21 February 1937) is King of Norway. He acceded to the throne on 17 January 1991. Harald was the third child and only son of King Olav V of Norway and Princess Märtha of Sweden. He was second in the li ...

at a ceremony in Oslo.

Personal life

In 1951, the

Massachusetts Institute of Technology The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the ...

(MIT) hired Nash as a

C. L. E. Moore instructor The job title of C. L. E. Moore instructor is given by the Math Department at Massachusetts Institute of Technology to recent math Ph.D.s hired for their promise in pure mathematics research. The instructors are expected to do both teaching and rese ...

in the mathematics faculty. About a year later, Nash began a relationship with Eleanor Stier, a nurse he met while admitted as a patient. They had a son, John David Stier, but Nash left Stier when she told him of her pregnancy. The film based on Nash's life, ''A Beautiful Mind'', was criticized during the run-up to the 2002 Oscars for omitting this aspect of his life. He was said to have abandoned her based on her social status, which he thought to have been beneath his. In

Santa Monica, California Santa Monica (; Spanish: ''Santa Mónica'') is a city in Los Angeles County, situated along Santa Monica Bay on California's South Coast. Santa Monica's 2020 U.S. Census population was 93,076. Santa Monica is a popular resort town, owing t ...

, in 1954, while in his twenties, Nash was arrested for indecent exposure in a sting operation targeting gay men. Although the charges were dropped, he was stripped of his top-secret

security clearance A security clearance is a status granted to individuals allowing them access to classified information (state or organizational secrets) or to restricted areas, after completion of a thorough background check. The term "security clearance" is ...

and fired from RAND Corporation, where he had worked as a consultant. Not long after breaking up with Stier, Nash met Alicia Lardé Lopez-Harrison, a

naturalized U.S. citizen Citizenship of the United States is a legal status that entails Americans with specific rights, duties, protections, and benefits in the United States. It serves as a foundation of fundamental rights derived from and protected by the Constit ...

from El Salvador. Lardé graduated from

, having majored in physics. They married in February 1957. Although Nash was an atheist, Nasar (2011), Chapter 17: Bad Boys, p. 143: "In this circle, Nash learned to make a virtue of necessity, styling himself self-consciously as a "free thinker." He announced that he was an atheist." the ceremony was performed in an Episcopal church. In 1958, Nash was appointed to a tenured position at MIT, and his first signs of mental illness soon became evident. He resigned his position at MIT in the spring of 1959. His son, John Charles Martin Nash, was born a few months later. The child was not named for a year because Alicia felt that Nash should have a say in choosing the name. Due to the stress of dealing with his illness, Nash and Lardé divorced in 1963. After his final hospital discharge in 1970, Nash lived in Lardé's house as a

boarder A boarder may be a person who: *snowboards *skateboards *bodyboards * surfs *stays at a boarding house *attends a boarding school *takes part in a boarding attack The Boarder may also refer to: * ''The Boarder'' (1953 film), a 1953 Soviet drama ...

. This stability seemed to help him, and he learned how to consciously discard his paranoid

delusion A delusion is a false fixed belief that is not amenable to change in light of conflicting evidence. As a pathology, it is distinct from a belief based on false or incomplete information, confabulation, dogma, illusion, hallucination, or som ...

s.David Goodstein, 'Mathematics to Madness, and Back'
''The New York Times'', June 11, 1998 Princeton allowed him to audit classes. He continued to work on mathematics and was eventually allowed to teach again. In the 1990s, Lardé and Nash resumed their relationship, remarrying in 2001. John Charles Martin Nash earned a PhD in mathematics from

Rutgers University Rutgers University (; RU), officially Rutgers, The State University of New Jersey, is a public land-grant research university consisting of four campuses in New Jersey. Chartered in 1766, Rutgers was originally called Queen's College, and was ...

and was diagnosed with

as an adult.

Death

On May 23, 2015, Nash and his wife died in a car accident on the New Jersey Turnpike near Exit 8A in Monroe Township, NJ. After a visit to Norway, where Nash had received the Abel Prize, they had made arrangements to be picked up by a limo at Newark Airport. But because of a change in flight plans at the last minute they arrived five hours earlier, and decided to take a taxi instead. Their taxicab driver, Tarek Girgis, lost control of the vehicle and struck a guardrail. Both passengers were ejected from the car upon impact. State police revealed that it appeared neither passenger was wearing a seatbelt at the time of the crash. At the time of his death, the 86-year-old Nash was a longtime resident of New Jersey. He was survived by two sons, John Charles Martin Nash, who lived with his parents at the time of their death, and elder child John Stier. Following his death, obituaries appeared in scientific and popular media throughout the world. In addition to their obituary for Nash, ''

The New York Times ''The New York Times'' (''the Times'', ''NYT'', or the Gray Lady) is a daily newspaper based in New York City with a worldwide readership reported in 2020 to comprise a declining 840,000 paid print subscribers, and a growing 6 million paid d ...

'' published an article containing quotes from Nash that had been assembled from media and other published sources. The quotes consisted of Nash's reflections on his life and achievements.

Legacy

At Princeton in the 1970s, Nash became known as "The Phantom of Fine Hall" (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 Rebecca Newberger Goldstein (born February 23, 1950) is an American philosopher, novelist, and public intellectual. She has written ten books, both fiction and non-fiction. She holds a Ph.D. in philosophy of science from Princeton University, and ...

. Sylvia Nasar's biography of Nash, '' A Beautiful Mind'', was published in 1998. A film by the same name was released in 2001, directed by

with Russell Crowe playing Nash; it won four

Academy Awards The Academy Awards, better known as the Oscars, are awards for artistic and technical merit for the American and international film industry. The awards are regarded by many as the most prestigious, significant awards in the entertainment ind ...

, including

Best Picture This is a list of categories of awards commonly awarded through organizations that bestow film awards, including those presented by various film, festivals, and people's awards. Best Actor/Best Actress *See Best Actor#Film awards, Best Actress#F ...

. For his performance as Nash, Crowe won the

Golden Globe Award for Best Actor – Motion Picture Drama The Golden Globe Award for Best Actor in a Motion Picture – Drama is a Golden Globe Award that was first awarded by the Hollywood Foreign Press Association as a separate category in 1951. Previously, there was a single award for "Best Actor i ...

and the BAFTA Award for Best Actor. Crowe was also nominated for the Academy Award for Best Actor for his performance as Nash at the

74th Academy Awards The 74th Academy Awards ceremony, presented by the Academy of Motion Picture Arts and Sciences (AMPAS), took place on March 24, 2002, at the Kodak Theatre in Hollywood, Los Angeles. During the ceremony, AMPAS presented Academy Awards (commonly ...

Awards

* 1978 – INFORMS John von Neumann Theory Prize (with Carlton Lemke) "for their outstanding contributions to the theory of games" * 1994 –

(with

and

) "for their pioneering analysis of equilibria in the theory of non-cooperative games" * 1999 – Leroy P. Steele Prize for Seminal Contribution to Research for his 1956 paper "The imbedding problem for Riemannian manifolds" * 2002 class of

Fellow A fellow is a concept whose exact meaning depends on context. In learned or professional societies, it refers to a privileged member who is specially elected in recognition of their work and achievements. Within the context of higher education ...

s of the

Institute for Operations Research and the Management Sciences The Institute for Operations Research and the Management Sciences (INFORMS) is an international society for practitioners in the fields of operations research (O.R.), management science, and analytics. It was established in 1995 with the merger o ...

* 2010 –

Double Helix Medal The Double Helix Medal has been awarded annually since 2006 by Cold Spring Harbor Laboratory (CSHL) to individuals who have positively impacted human health by raising awareness and funds for biomedical research. At the inaugural dinner, Muhammad ...

* 2015 – Abel Prize (with

) "for striking and seminal contributions to the theory of nonlinear partial differential equations and its applications to geometric analysis"

Documentaries and interviews

* * * * (, ) *

Publication list

* * * * * * * * * * * * * * * * * * * * * * * Four of Nash's game-theoretic papers and three of his pure mathematics papers were collected in the following: *

References

Bibliography

* * * *

External links

Home Page of John F. Nash Jr. at Princeton
*

2002 '' Slate'' article by Robert Wright, about Nash's work and world government
NSA releases Nash Encryption Machine plans
to

National Cryptologic Museum The National Cryptologic Museum (NCM) is an American museum of cryptologic history that is affiliated with the National Security Agency (NSA). The first public museum in the U.S. Intelligence Community, NCM is located in the former Colony Sev ...

for public viewing, 2012 *
Nash, John (1928–2015) , Rare Books and Special Collections
from Princeton's Mudd Library, including a copy o
his dissertation
(PDF)
Biography of John Forbes Nash Jr.
from the Institute for Operations Research and the Management Sciences * {{DEFAULTSORT:Nash, John Forbes Jr. 1928 births 2015 deaths 20th-century American mathematicians Abel Prize laureates American atheists American Nobel laureates Board game designers Carnegie Mellon University alumni Institute for Advanced Study visiting scholars Differential geometers Fellows of the American Mathematical Society Fellows of the Econometric Society Fellows of the Institute for Operations Research and the Management Sciences Game theorists John von Neumann Theory Prize winners Massachusetts Institute of Technology School of Science faculty Members of the United States National Academy of Sciences Nobel laureates in Economics PDE theorists People from Bluefield, West Virginia People from West Windsor, New Jersey People with schizophrenia Road incident deaths in New Jersey Princeton University alumni Princeton University faculty Mathematicians from West Virginia Mathematicians from New Jersey McLean Hospital patients Members of the American Philosophical Society