## VI Arnold

### Mathematician and teacher renowned for his incisive solutions

VI Arnold, who has died of pancreatitis aged 72, was a giant of mathematics. A prolific creator and an inspiring teacher, he had one of the sharpest and most imaginative minds in the business. His research was renowned for its clarity and incisiveness. He made major contributions to dynamical systems and singularity theory, with ramifications in pure and applied mathematics: algebra, topology, algebraic geometry, celestial mechanics, fluid dynamics and optics.Vladimir Igorevich Arnold was born in Odessa and entered the faculty of mathematics at Moscow State University as a student in 1954. Three years later, still in his teens, he solved Hilbert's 13th Problem, one of a series first stated by David Hilbert in 1900. He got his PhD in 1961 and was made a professor at the university in 1965, remaining there until 1986, when he moved to the Steklov Institute, in Moscow. From 1993 until 2005 he split his time between there and the Université Paris Dauphine, in France.

In classical mechanics, where energy is conserved, solutions sometimes have an unusually regular form, being combinations of periodic motions with different periods. For example, the moon repeats its orbit around the Earth once a month, and the Earth orbits the sun in a year, so the moon's motion relative to the sun is a complicated combination of two simpler periodic motions. Andrei Kolmogorov, Arnold's PhD supervisor, had compiled evidence that when the system is disturbed by a small perturbation, such as the influence of another planet, some of these motions persist.

One of Arnold's achievements was to provide rigorous proof of Kolmogorov's result and extend it to more complex cases, such as the solar system. This was remarkable, because many had thought that such solutions of simplified models would not survive when the models became more realistic.

The result is now known as KAM theory, after Kolmogorov, Arnold and Jürgen Moser, who developed important related ideas. KAM theory establishes that many systems of classical mechanics have a plentiful supply of states that resemble a combination of periodic oscillations with several different frequencies. For instance, the tides have component oscillations with frequencies of half a lunar day, a lunar day, half a lunar month, a lunar month, half a year, a year, and so on. Oscillations of this kind are stable: they remain within fixed bounds, and do not die out or explode. This pointed towards a potential application to a major problem -- but there was a snag.

One of the big questions about the solar system is its stability. Will the planets continue to orbit the sun indefinitely, close to their present orbits? Or will there be some dramatic change, such as two planets colliding, or one being flung into the depths of interstellar space? This question had taxed many great mathematicians in the past. Isaac Newton discovered that the orbit of a planet round the sun is an ellipse, which is stable, but the calculation ignores the effects of any other bodies. However, the solar system contains eight planets -- not counting Pluto, which is no longer considered a planet -- and numerous other smaller bodies. Every one of these bodies exerts a gravitational force on every other, in the very particular way specified by Newtonian gravity.

Unfortunately, Kolmogorov's theorem does not apply to the solar system, because Newtonian gravity does not obey the necessary technical conditions. Arnold found a way around this obstacle, and proved stability for an idealised solar system in which all planets have small masses and initially move in circular orbits in the plane. More precisely, he proved this for two planets, and made progress on an arbitrary number of planets. The proof was later completed by Michel Herman. The masses of real planets are too large for the theorem to apply, and their orbits are neither circles nor in the same plane. Nevertheless, Arnold's ideas had a profound philosophical impact.

Arnold was the first to acknowledge the limitations of his result. He constructed an example to show that other trajectories can wander in the gaps between the stable ones, making arbitrarily large excursions. Boris Chirikov christened this behaviour "Arnold diffusion". Numerical computations by Jacques Laskar have since made it plausible that the solar system is in fact unstable, with deviations growing by a factor of about 10 every 10m years. Collisions, or the ejection of a planet, are possible, though neither is likely to occur for at least a billion years. A rigorous proof is still lacking.

Arnold was also an early contributor to what is popularly known as "chaos theory", the branch of dynamics that deals with apparently random solutions to non-random equations. In parallel with the French topologist René Thom, he proposed a simple example of a dynamical system that appears to be random when observed to any given level of precision. Only with infinitely precise observations can it be seen not to be random. This example is known as "Arnold's cat", because he illustrated the dynamics by showing how a drawing of a cat evolves.

He applied this example to the Earth's magnetic field, which is thought to be produced by convection currents in its liquid iron core. There is a major theoretical issue here: can the flow of a conducting fluid in three-dimensional space generate a magnetic field? Arnold's example does this, but it is not realisable in ordinary three-dimensional space. However, a simple modification produces a more realistic flow that may well possess the same dynamo property.

The second great area of Arnold's creativity is singularity theory. This is the study of the typical ways that one space can be folded into another. Imagine laying a tablecloth on a table. If it is laid flat, then each point of the tablecloth lies above a distinct point on the table. If instead you fold the tablecloth once, then the points on the table fall into three types. Those under the tablecloth but not along its folded edge lie beneath two distinct points of the tablecloth: one on its upper layer, the other on its lower layer. Those along its folded edge lie under a single point of the tablecloth. The rest lie beneath no points of the tablecloth. So the edge of the tablecloth determines a curve of "fold points", at which the correspondence between table and tablecloth changes from two-to-one, through one-to-one, to zero-to-one. As the point on the table crosses this curve, the two points of the tablecloth that lie above it merge, and then annihilate each other. With typical humour, Arnold referred to such structural changes as perestroika.

There are other ways to lay a mathematical tablecloth on top of a table -- that is, to map a plane into a plane. They include multiple folds, but also pleats, where a triangular region is created in which three points of the tablecloth lie above each point on the table. The mathematical term for such a pleat is "cusp". Arnold built on results of the American topologist Hassler Whitney and Thom to produce an all-encompassing theory of the typical geometry of such folds, cusps and more complex "singularities", and how they join together.

The resulting theory has a broad range of applications covering areas as disparate as optics, mechanics and cosmology. One of his most profound contributions was to prove an intimate connection between the classifications of singularities of smooth mappings and that of Lie groups analogues of rotations in spaces of many dimensions. An example of his results is an explanation of the four-pointed star shape that appears when the stem of a wineglass focuses the sun on to a table.

In the field of fluid dynamics, particularly the ideal case of incompressible fluids with no viscosity, Arnold adapted the ideas of Peter Dirichlet and Karl Weierstrass to prove the stability of various flow patterns, a method that is now widely used. He realised that the flow-lines in a fluid can be chaotic, long before the mainstream fluid dynamics community recognised this. He also spotted a profound analogy between the equations of motion for fluids, and those for a rotating rigid body. This analogy made it clear that most fluid flows are unstable. Extrapolating from this ideal case, he argued that this places limits on weather forecasting.

Arnold could be blunt and outspoken, and he spoke forcefully -- though in measured tones -- about anything with which he disagreed, such as the trend to abstract formalisation of mathematics. In a small but elegant book, he criticised "catastrophe theory", a spin-off from singularity theory advanced by Thom and Christopher Zeeman, which adventured from its firm mathematical basis to attempt to explain biological phenomena such as morphogenesis and social phenomena such as prison riots.

Though Arnold received many honours, one that eluded him was the Fields medal, long considered the mathematical analogue of the Nobel prize. He was nominated in 1974, but support for him by the Soviet Union was suddenly withdrawn. Arnold thought this was perhaps because he had signed a letter protesting at the imprisonment of a perfectly healthy Soviet mathematician in a psychiatric hospital. He never shied away from saying what he thought was right.

He is survived by his wife, Eleanora, a brother and sister, two sons, four grandchildren and two great-grandchildren.

**Robert Sinclair MacKay and Ian Stewart**

*Vladimir Igorevich Arnold, mathematician, born *12 *June* 1937; *died* 3 *June* 2010

20 September 2010 © Guardian Newspapers Limited