Vladimir ArnoldVladimir Arnold, who died on June 3 aged 72, was one of the great mathematicians of the 20th century, whose work helped explain why the systems around us, at any scale from particles to planets, work as they do.
He found that small changes had an immense impact on outcomes. His study of particles helped him determined how fluids flow. That understanding was applied to weather systems, and explained why long range forecasting was effectively impossible because small events could have dramatic, unforeseen, consequences.
This idea could be taken to an even larger scale -- that of our solar system. Arnold, together with Andrei Kolmogorov (his adviser at university) and Jurgen Moser, made fundamental contributions to the study of stability in dynamic systems, exemplified by the motion of the planets around the sun.
The genesis of the problem can be traced back to Newton. He was able to solve the equations that determine the motion of two bodies (say, the sun and the earth) as they interact with each other through gravitational force. However, when he added a third body (say, the moon), he was unable to solve the corresponding equations for of all three bodies.
This is the famous 3-body problem, in which the motion of each body seemed impossible to predict. The Kolmogorov-Arnold-Moser (KAM) theory is a body of work that addressed this problem, and laid down proofs for a certain set of circumstances.
Arnold had a geometric approach to topics such as ordinary differential equations which gave his students an intuitive understanding of the problem at hand. "Arnold's Cat Map", for example, displays the properties that many chaotic systems show through transformation. An image, in this case of a cuddly kitten, is stretched and then wrapped to restore the original dimensions (but not the original image). If this done enough times, Arnold found that the original image is also restored.
He posed problems that could be solved in physical terms. It was he who first raised the question of whether a convex and homogeneous body, with one unstable and one stable equilibrium, could exist in three dimensions. Such thoughts spawned the "Gomboc" which is something a bit like a roly-poly toy.
"Our brain has two halves," Arnold noted. "One is responsible for the multiplication of polynomials and languages, and the other is responsible for the orientation of figures in space and all the things that are important in real life. Mathematics becomes geometry when you use both halves."
Vladimir Igorevich Arnold was born on June 12 1937 in Odessa, then part of the USSR and now in Ukraine. Several generations of his family had been mathematicians. He entered Moscow State University in 1954 and took his first degree in 1959. His Candidate's Degree (equivalent to a PhD.) was awarded in 1961 for a thesis which solved a version of Hilbert's 13th Problem.
In 1900 the German mathematician, David Hilbert, had presented a list of 23 then unsolved problems to a Congress in Paris. One of these, the 13th, involved solving the general seventh degree equation using functions of two variables. By solving the problem, Arnold secured his reputation while still in his early twenties.
In 1965 Arnold became Professor of Mechanics at Moscow State University, where he remained until 1986, when he took up a position at the Steklov Institute of Mathematics in Moscow. Having upset the Soviet authorities in the 1960s, he was unable to leave the Soviet Union from that time until the late 1980s. This enforced stability perhaps contributed to his belief that American mathematicians spend their time travelling to conferences whereas the Russians sit at home, working hard to prove fundamental theorems destined forever to remain the cornerstones of mathematics.
Indeed, Arnold's relationship with authorities was so bad that, in 1974, the Soviet Union opposed the award to him of the Fields Medal, the foremost recognition of work in mathematics. This resulted in Arnold being one of the most prominent mathematicians never to receive the prize, often compared with the Nobel. He did, however, receive virtually every other international mathematics prize as well as numerous doctorates and honorary degrees from around the world. He was awarded the Wolf prize in 2001 "for his deep and influential work in a multitude of areas of mathematics, including dynamical systems, differential equations and singularity theory".
Despite his views on globetrotting American academics, Arnold himself began to travel a great deal after his own ban on leaving the USSR was lifted. In 1993 he was appointed Professor at the University Paris-Dauphine in France and divided his time between Paris and Moscow. He held this position until 2005.
Arnold was also an inspired teacher of mathematics. He had strong views on the importance of the applications of maths and had no time for the abstract approach typified by the Bourbaki school in France. As he said: "Mathematics is part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap."
Accordingly, his lectures were not dry. Difficult modern theories became quite clear and simple in his exposition and he had a special gift for finding new and beautiful problems to interest and involve young researchers.
He started one such seminar by saying: "There is a general principle that a stupid man can ask such questions to which one hundred wise men would not be able to answer. In accordance with this principle I shall formulate some problems."
Arnold was a keen sportsman from childhood. Even in recent years he gave seminars after completing a 50 kilometre bicycle ride. He was an Alpinist and liked to swim in icy ponds in the winter. He insisted that sports were essential for thinking and that often his best work was done on the move.
Arnold is survived by his wife, Voronina Elionora Aleksandrova, whom he married in 1976, and by his two sons.
12 Jul 2010 © Telegraph Group Limited 2010.