John Forbes Nash Jr.

Lore & Background

He attended Carnegie Institute of Technology on a George Westinghouse Scholarship, initially studying chemical engineering before switching to mathematics. As a graduate student, Nash introduced the Nash equilibrium and Nash bargaining solution, foundational to game theory. In the 1950s, he proved the Nash embedding theorems by solving nonlinear partial differential equations, work that introduced a preliminary form of the Nash–Moser theorem. With Ennio De Giorgi, he developed the De Giorgi–Nash theorem on the smoothness of solutions to elliptic and parabolic partial differential equations, resolving Hilbert's nineteenth problem.

Reader's Guide

John Nash's significance lies in his transformative contributions across multiple mathematical fields. His Nash embedding theorems, which proved that any Riemannian manifold can be isometrically embedded in Euclidean space, are considered among the major achievements of 20th-century mathematics, with Mikhael Gromov calling the second theorem 'one of the main achievements of mathematics of the 20th century.' The De Giorgi–Nash theorem resolved Hilbert's nineteenth problem, a long-standing open question in the calculus of variations, and paved the way for systematic understanding of partial differential equations. Nash's work also influenced fluid mechanics through convex integration and modern cryptography through declassified letters. His legacy endures in the fundamental concepts and theorems that bear his name.

Did You Know?

• Nash's second embedding theorem was called 'one of the main achievements of mathematics of the 20th century' by Mikhael Gromov.
• Nash and Ennio De Giorgi independently developed the De Giorgi–Nash theorem, which resolved Hilbert's nineteenth problem.