Nash had arrived at Princeton University, New Jersey, in 1948 to study for a postgraduate degree in mathematics, bearing a laconic one-line recommendation from his previous professor, Richard Duffin: "This man is a genius." He proved his genius within two years by publishing what is surely the shortest paper ever to win its author a Nobel prize. Called *Equilibrium Points in N-Person Games*, it was less than a page long and contained just 317 words.

It was a major contribution to the burgeoning field of game theory, whose foundations had been laid by the mathematician John von Neumann and the economist Oskar Morgenstern at the Princeton Institute for Advanced Study in their Theory of Games and Economic Behavior (1944). With Nash's insight, game theory provides a focus for understanding the roots of many problems in conflict and the failure of co-operation that we face today.

Nash himself, however, did not regard his contribution particularly highly. He was more concerned with tackling basic problems across the whole range of mathematics. Even as an undergraduate he had produced an independent proof of Brouwer's fixed-point theorem -- the theorem that tells us that, no matter how much we stir a cup of coffee, there will always be one small bit that is just where it was before we started stirring.

In 1954 and 1956 he produced the two remarkable Nash embedding theorems, which are difficult to explain to non-mathematicians -- they prove that "every Riemannian manifold can be isometrically embedded into some Euclidean space" -- but which provided the basis for much subsequent mathematics. One very simplified way to illustrate their main point is through the proposition that no matter how you bend a piece of paper with lines drawn on it, the lines will always have the same length. It sounds intuitively obvious, but the proof was difficult, and the theorem has led to many counterintuitive conclusions.

The mental illness that incapacitated Nash and eventually prompted *A Beautiful Mind* began to make itself felt in 1959. He could be quite funny about it in later years, and told a New Scientist interviewer in 2004: "Mathematicians are comparatively sane as a group. It is the people who study logic that are not so sane." But the reality was far from funny. His diagnosed paranoid schizophrenia led to the breakup of his marriage -- although he remarried his wife Alicia (who died with him in the taxi cab crash) in 2001 -- the loss of his job, and nearly the loss of his Nobel prize when members of the selection committee were worried that he might never be able to pursue serious research again.

But carry on he did. In between bouts of hospitalisation, with gradual recovery after many severe episodes in the 1970s, 80s and early 90s, Nash continued to make contributions to many areas of mathematics, especially in the solution of partial differential equations, which describe how several factors affect each other when all are changing simultaneously. Such equations are basic tools for engineers and for people developing computer models of economies and societies, but they are notoriously difficult to solve.

For his contributions Nash was awarded the Abel prize by the Norwegian Academy of Arts and Letters on 19 May. He shared the prize with Louis Nirenberg, and talked of his plans for work in the fields of cosmology and general relativity: when he first joined Princeton, he had reputedly demanded an interview with Albert Einstein so that he could explain where the general theory of relativity was wrong.

Nash was born in Bluefield, West Virginia. His father, John Forbes Nash Sr, was an electrical engineer, and his mother, Margaret Virginia (nee Martin), had been a teacher. After taking advanced mathematics courses at a local community college during his final year of high school, he attended the Carnegie Institute of Technology (now Carnegie Mellon University), initially majoring in chemical engineering. He switched to chemistry, and then to mathematics, graduating in 1948 with bachelor's and master's degrees in mathematics, after which he accepted a fellowship to study for a doctorate at Princeton.

On completion of his thesis -- just 27 pages long -- he was hired by Massachusetts Institute of Technology (MIT), at first as an instructor in the mathematics faculty, and then in 1958 as a tenured professor. He began a relationship with a nurse, Eleanor Stier, and in 1953 they had a son, John Stier. In 1957 he married Alicia Lopez-Harrison de Lardé. He was also a consultant for the RAND corporation from 1950 to 1954, but lost his position, even though all charges were dropped, after he had been arrested for indecent exposure in a police entrapment operation in Santa Monica, California.

Nash's illness led to his resignation from his position at MIT in the spring of 1959, and Alicia had him admitted to McLean hospital, in Belmont, Massachusetts, for treatment. Their son, John Nash, who also became a mathematician, was born that May.

The stress of Nash's illness took its toll, and he and his wife were divorced in 1963. After various periods of hospitalisation he was finally discharged in 1970, but continued to struggle with paranoid delusions over the next two decades. In the New Scientist interview, he said: "I was a long way into mental illness before I heard any voices. Ultimately I realised I am generating these voices in my own mind: this is dreaming, this is not communication. This is coming from an internal source, not from the cosmos. And simply to understand that is to escape from the thing in principle. After understanding that, the voices died out." Of his portrayal in film he said: "It's not me, but Russell Crowe plays the role well."

In addition to his Nobel and Abel prizes, Nash was awarded the John von Neumann Theory prize in 1978, and the Leroy P Steele prize in 1999.

He is survived by his two sons and a sister, Martha.

**Len Fisher**

*John Forbes Nash, mathematician, born* 13 *June* 1928; *died* 23 *May* 2015

Monday 25 May 2015 © Guardian News and Media Limited