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John Nash’s unique approach produced quantum leaps in economics and maths

John Nash’s unique approach produced quantum leaps in economics and maths

THE American mathematician John Nash, who was killed on Saturday night in a car crash, was in Oslo five days ago to receive the Abel prize from the king of Norway. The half a million pounds Abel — which he shared with Louis Nirenberg — is considered a kind of maths version of the Nobel Prize, which has no category for mathematics.

And yet, Nash is also a winner of the Nobel Prize, the only person to share both accolades. “I must be an honorary Scandinavian,” he joked in March during the press conference that announced this year’s Abel laureates.

Nash is most famous for his research into game theory, the maths of decision-making and strategy, since it was this work that led to his being awarded the 1994 Nobel in economics. His fame also came from the 2001 Oscar-winning film A Beautiful Mind, in which he was played by Russell Crowe. The film, which turned him into probably the best-known mathematician in the world, was based on the superb biography by Sylvia Nasar and charts his early career and then the struggle with schizophrenia that dominated most of his adult life.

Within the mathematics community, however, the work for which Nash was most admired — and for which he won the Abel — was not the game theory research but his advances in pure mathematics, notably geometry and partial differential equations.

The mathematician Mikhail Gromov once said: “What [Nash] has done in geometry is, from my point of view, incomparably greater than what he has done in economics, by many orders of magnitude. It was an incredible change in attitude of how you think.”

Nash’s achievements in mathematics were striking not only because he proved deep and important results, but also because his career lasted only a decade before he was lost to mental illness.

Nash was born in 1928 in a small, remote town in West Virginia. His father was an electrical engineer and his mother a schoolteacher. He was an undergraduate at Carnegie Institute of Technology (now Carnegie Mellon University) in Pittsburgh and then did his graduate studies at Princeton, New Jersey. His PhD thesis, Non-Cooperative Games, is one of the foundational texts of game theory. It introduced the concept of an equilibrium for non-cooperative games, the “Nash equilibrium”, which eventually led to his economics Nobel Prize.

Yet his mathematical interests soon lay elsewhere. He described his first breakthrough in pure mathematics, in his early 20s, as “a nice discovery relating to manifolds and real algebraic varieties”. His peers already recognised the result as an important and remarkable work.

In 1951, Nash left Princeton for MIT. Here, he became interested in the problems of “ isometric embedding” , which ask whether it is possible to embed abstractly defined geometries into real-world geometries in such a way that distances are maintained. Nash’s two embedding theorems are considered classics, providing some of the deepest mathematical insights of the last century.

This work on embeddings led him to partial differential equations, which are equations involving flux and rates of change. He devised a way to solve a type of partial differential equation that hitherto had been considered impossible. His technique, later modified by Jurgen Moser, is now known as the Nash-Moser theorem.

In the early 1950s, Nash worked during the summers for the RAND Corporation, a civilian think tank funded by the military in Santa Monica, California. Here, his work on game theory found applications in United States military and diplomatic strategy.

Perhaps Nash’s greatest mathematical work came from studying a mathematical puzzle that had been suggested to him by Louis Nirenberg. It concerned a major open problem concerning elliptic partial differential equations. Within a few months, Nash had solved the problem. It is thought that his work would have won him the Fields Medal — the most prestigious prize in maths, open only to those under 40 — had it not been solved at the same time by Italian mathematician Ennio De Giorgi. The men used different methods, and were not aware of each other’s work — the result is known as the Nash-De Giorgi theorem.

One of the many amazing aspects of Nash’s career was that he was not a specialist. Unlike almost all top mathematicians now, he worked on his own, and relished attacking famous open problems, often coming up with completely new ways of thinking. Louis Nirenberg once said: “About 20 years ago somebody asked me, “Are there any mathematicians you would consider as geniuses?” I said, “I can think of one, and that’s John Nash … He had a remarkable mind. He thought about things differently from other people.”

In 1959, Nash began to suffer from delusions and extreme paranoia. For the next 40 years or so he was only able to do serious mathematical research in brief periods of lucidity. Remarkably, however, he gradually improved and his mental state had recovered by the time he won the Nobel in 1994.

Nash showed such resolve and stamina in his mathematical work and in recovering from his mental illness, that his death in a taxi crash on the New Jersey turnpike seems all the more pointless and tragic.

—By arrangement with the Guardian

Published in Dawn, May 26th, 2015

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