John Forbes Nash Jr.

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.

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's parents pursued opportunities to supplement their son's education, and arranged for him to take advanced mathematics courses at a local community college (Bluefield College) 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 a B.S. and M.S. 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 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.

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.

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 1949, while still a graduate student, Nash found a new result in the mathematical field of real algebraic geometry.

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 sixty years.

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.

In the early 1950s, Nash carried out research on a number of related concepts in game theory, including the theory of cooperative games.

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.

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 improved, allowing him to return to academic work by the mid-1980s.

Nash and fellow game theorists John Harsanyi and Reinhard Selten were awarded the 1994 Nobel Memorial Prize in Economics.

For his work, Nash was one of the recipients of the Nobel Memorial Prize in Economic Sciences in 1994.

Nash was biographed in Sylvia Nasar's 1998 book A Beautiful Mind, and his struggles with his illness and his recovery became the basis for a film of the same name directed by Ron Howard, in which Nash was portrayed by Russell Crowe.

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.

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 Nash equilibrium and the Nash bargaining solution) which are now considered central to game theory and its applications in various sciences.