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 game theory, real algebraic geometry, differential geometry, and partial differential equations. Nash and fellow game theorists John Harsanyi and Reinhard Selten were awarded the 1994 Nobel Prize in Economics. In 2015, Louis Nirenberg and he were awarded the Abel Prize for their contributions to the field of partial differential equations.
As a graduate student in the Princeton University Department of Mathematics, Nash introduced a number of concepts (including the Nash equilibrium and the Nash bargaining solution), 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. This work, also introducing a preliminary form of the Nash–Moser theorem, was later recognized by the American Mathematical Society 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 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 60 years.
In 1959, Nash began showing signs of mental illness and spent several years at psychiatric hospitals being treated for schizophrenia. After 1970, his condition slowly improved, allowing him to return to academic work by the mid-1980s.
Early life and education
John Forbes Nash Jr. was born on June 13, 1928, in Bluefield, West Virginia. 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 was also accepted at Harvard University, along with the University of Chicago and the University of Michigan. However, the chairman of the mathematics department at Princeton, Solomon Lefschetz, offered him the John S. Kennedy<!-- NOTE: not John F. Kennedy--> fellowship, convincing Nash that Princeton valued him more. Further, he considered Princeton more favorably because of its proximity to his family in Bluefield.
Research contributions
thumb|upright|Nash in November 2006 at a [[game theory conference in Cologne, Germany]]
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 real algebraic geometry. As a postdoctoral fellow at MIT, 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 of partial differential equations. 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 declassified letters written by Nash in the 1950s, in which he had proposed a new encryption–decryption machine. The letters show that Nash had anticipated many concepts of modern cryptography, which are based on computational hardness.
Game theory
Nash earned a PhD in 1950 with a 28-page dissertation on noncooperative 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 noncooperative games. A version of his thesis was published a year later in the Annals of Mathematics. 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 in 1994.
Real algebraic geometry
In 1949, while still a graduate student, Nash found a new result in the mathematical field of real algebraic geometry. 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 Annals of Mathematics. It had been well-known since the 1930s that every closed smooth manifold is diffeomorphic to the zero set of some collection of smooth functions on Euclidean space. In his work, Nash proved that those smooth functions can be taken to be polynomials. 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 and Barry Mazur to the study of dynamical systems, by combining Nash's polynomial approximation together with Bézout's theorem.
Differential geometry
During his postdoctoral position at MIT, Nash was eager to find high-profile mathematical problems to study. From Warren Ambrose, a differential geometer, he learned about the conjecture that any Riemannian manifold is 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 of the main achievements of mathematics of the 20th 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 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 mathematical analysis. Starting with work of Camillo De Lellis and László Székelyhidi, the ideas of Nash's proof were applied for various constructions of turbulent solutions of the Euler equations in fluid mechanics. In the 1970s, Mikhael Gromov developed Nash's ideas into the general framework of convex integration,
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 along which extra regularity is continually injected, is regarded as a fundamentally novel technique in mathematical analysis. 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".
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, Jürgen Moser 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 Nash–Moser theorem). De Giorgi and Moser's methods became particularly influential over the next several years, through their developments in the works of Olga Ladyzhenskaya, James Serrin, and Neil Trudinger, among others. Their work, based primarily on the judicious choice of test functions in the weak formulation 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 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.
Mental illness
Although Nash's mental illness first began to manifest in the form of paranoia, his wife later described his behavior as erratic. Nash thought that all men who wore red ties were part of a "crypto-communist party" who were secretly conspiring against him. He mailed letters to embassies in Washington, D.C., declaring that he was establishing a government. Nash signed these letters "John Nash, Emperor of Antarctica", a position he believed he was in line to inherit.
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. In 1961, Nash was admitted to the New Jersey State Hospital at Trenton. Over the next nine years, he spent intervals of time in psychiatric hospitals, where he received both antipsychotic medications and insulin shock therapy.
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 antipsychotics. 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, Alicia 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. Lardé credits his recovery to maintaining "a quiet life" with social support. During his psychotic phase, Nash also referred to himself in the third person as "Johann von Nassau". 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.
Nash wrote in 1994: