John Nash won a Nobel for 28 pages, but mathematicians prize his geometry more
Most people know John Nash from A Beautiful Mind and from the equilibrium in game theory that earned him an economics Nobel. Yet the mathematician Mikhael Gromov called one of Nash's geometry theorems among the main achievements of twentieth-century mathematics, and said it was as impossible as the story of his life.
Nash was born in Bluefield, West Virginia, on June 13, 1928, the son of an electrical engineer and a former schoolteacher. His parents arranged college mathematics classes for him while he was still in high school. At Carnegie Tech he drifted from chemical engineering to chemistry to mathematics, and his recommendation letter for graduate school contained a single striking line: he is a mathematical genius. Princeton won him over with a fellowship and its nearness to home.
There he wrote a 28-page doctoral thesis on noncooperative games in 1950, defining the balance point now called the Nash equilibrium, in which no player gains by changing strategy alone. That work brought him a share of the 1994 Nobel Memorial Prize in Economic Sciences with John Harsanyi and Reinhard Selten. Still a student, he also proved that the smooth shapes mathematicians call manifolds can always be described using plain polynomials, a surprise given how much more flexible smooth functions usually are.
At MIT he went hunting for famous open problems. He showed that any curved space of a certain kind can be placed inside ordinary flat space without distorting distances, first in a rough form in 1953, then, after about a year and a half of grinding, in a smooth version whose method was so unusual that one colleague called it not just novel but very mysterious. With separate methods from Ennio De Giorgi, he also settled a question about the smoothness of solutions to a family of equations, resolving Hilbert's nineteenth problem after nearly 60 years.
In 1959 he began showing signs of schizophrenia and spent years in psychiatric hospitals. His condition improved slowly after 1970, and by the mid-1980s he was working again. Declassified letters released in 2011 showed he had anticipated modern cryptography's reliance on hard computations. In 2015 he shared the Abel Prize with Louis Nirenberg, shortly before his death that May.
Source: John Forbes Nash Jr.
