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 Memorial Prize in Economic Sciences. In 2015, he and Louis Nirenberg 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.
Nash's life was the subject of Sylvia Nasar's 1998 biographical book A Beautiful Mind, and his struggles with his illness and his recovery became the basis for a 2001 film of the same name directed by Ron Howard, in which Nash was portrayed by Russell Crowe.
Early life and education John Forbes Nash Jr. was born on June 13, 1928, in Bluefield, West Virginia. His father, 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. His younger sister Martha was born in 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 nearby Bluefield College (now Bluefield University) during his final year of high school. He attended 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 major and eventually, at the advice of his teacher John Lighton Synge, to mathematics. After graduating in 1948, with both bachelor of science and master of science degrees in mathematics, Nash accepted a fellowship to Princeton University, where he pursued further graduate studies in mathematics and sciences.
Nash's adviser and former Carnegie Tech professor Richard Duffin wrote a letter of recommendation for Nash's entrance to Princeton, stating, "He is a mathematical genius." Nash was also accepted at Harvard University, 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 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.Nasar (2002), pp. xvi–xix.
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 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, which has been (among other uses) applied by Stefan Müller and Vladimír Šverák 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 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". According to Gromov:
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 known as the Nash–Moser theorem. It has been extended and generalized by a number of other authors, among them Gromov, Richard Hamilton, Lars Hörmander, Jacob Schwartz, and Eduard Zehnder. 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:
Partial differential equations While spending time at the Courant Institute in New York City, Louis Nirenberg 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 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 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 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), which has been found useful in other contexts.