We learn at school how to solve polynomial equations of first and second degree, or linear and quadratic equations. In the early 16th century, Italian mathematicians discovered how to solve equations of third and fourth degree, or cubic and quartic equations. But no one could solve quintics; equations of the fifth degree. At least, not until Niels Henrik Abel appeared.
Abel (1802-1829) was a Norwegian mathematician whose most famous result was to prove the impossibility of solving equations of degrees greater than four in terms of explicit algebraic operations. This question was one of the outstanding open problems of his day. Abel made other significant mathematical advances, in elliptic functions and group theory, but he was largely unrecognised during his lifetime. It is said that Carl Friedrich Gauss, the greatest mathematician of the age, dismissed the paper that Abel sent to him as the valueless work of a crank.
In 2002 the Norwegian government established a fund of 200 million kroner to finance the award of a prize to commemorate Abel, one of the nation’s greatest mathematicians. The Abel Prize has been awarded annually since 2003. Recently, the prize was presented by King Harald V to John Forbes Nash and Louis Nirenberg. The citation accompanying the award was for “striking and seminal contributions to the theory of nonlinear partial differential equations and its applications to geometric analysis”.
The name of John Nash will be familiar to anyone who has watched the film A Beautiful Mind, which is the story of how a sparkling academic career unravelled as Nash began to suffer from paranoia and delusions. His mental state deteriorated and he spent years in psychiatric care. Gradually, with great support from his wife, Alicia, he recovered his mental equilibrium and returned to productive work.
Game theory stresses the idea of a zero-sum game, but most real-world interactions are more complicated. Nash showed that in a competitive situation, if both parties are motivated by self-interest with each acting to maximise his own benefits, there can still be opportunities for mutual gain. He presented his ideas in a doctoral thesis when he was just 21. His equilibrium model has had a profound influence on economic theory and in 1994 he shared a Nobel prize in economics for this work.
But Nash was one of the greatest mathematicians of the 20th century, and many mathematicians view his work on partial differential equations (PDEs) as more significant than his work on game theory. PDEs are used to describe the basic laws of physics. For example, the computer models used to forecast the weather are based on a system of PDEs. These equations are also important in chemistry, biology and other sciences. Nash and Nirenberg made major contributions to the theory of PDEs by solving fundamental problems and introducing deep and novel ideas.
For uncertain reasons, Alfred Nobel did not endow any prize for mathematics. The Abel Prize, comparable in prestige to the Nobel, is awarded “for outstanding scientific work in the field of mathematics”. The amount awarded is six million kroner, or about €700,000. The award of the prize to Nash was the crowning moment in his career, but tragedy soon followed. While they were returning home from Norway after a week of celebrations, Nash and his wife were killed when they were thrown from a taxi in a crash on the New Jersey Turnpike.
- Peter Lynch is emeritus professor at the school of mathematical sciences, University College Dublin. He blogs at thatsmaths.com