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 ==