## Rene Thom

### Creator of 'catastrophe theory' whose mathematical ventures into higher dimensions won him the Fields Medal

©Rene Thom, French mathematician, Fields Medallist, philosopher, and the creator of catastrophe theory, was born in Montbeliard on September 2, 1923. He died at Bures-sur-Yvette on October 25, 2002, aged 79.Rene Thom was the co-founder (with Hassler Whitney) of the two new mathematical fields of differential topology and singularity theory. During the 1950s he invented the key tools of these fields, created cobordism theory and classified generic singularities of smooth maps. Cobordism is a way of classifying manifolds, the higher dimensional analogues of spheres and other curved surfaces. When one manifold is mapped into another there are points where the first is unavoidably crinkled up, and these points are called singularities. Manifolds and singularities are fundamental to geometry, and to the understanding of all branches of pure and applied mathematics that use equations.

For this work Thom was awarded in 1958, at the age of 35, a Fields Medal, the equivalent in mathematics of a Nobel Prize. Three years later he was appointed to a permanent professorship at the newly created Institut des Hautes Etudes Scientifiques at Bures-sur-Yvette near Paris, where he remained for the rest of his life.

Thom said that the Fields Medal had brought him the freedom to choose what research he wanted to do, and for him that was essential. He began to take the whole of science as his canvas. He was not a theoretical or experimental scientist, in the sense of designing experiments and predicting results, but rather a philosopher of science, writing about the long-term future developments in the sciences that needed to occur.

In particular he created catastrophe theory to explain how discrete forms can emerge out of homogeneity, and how continuous causes can give rise to discontinuous effects, such as boiling, buckling, capsizing, or the changes observed in a growing embryo. Normally if a large, bad, unexpected discontinuity happens then it is liable to be called a catastrophe, and that suggested the name of the theory.

The underlying mathematics of catastrophe theory concerns dynamical systems with parameters: if the parameters are changed gradually then the equilibrium will respond by changing gradually but may then suddenly lose its stability, causing the system to jump into another equilibrium. Graphically, one parameter can be represented by an axis, and the equilibriums by a curve sitting over this axis. If this curve folds over then the jump will occur at the fold point. If there are two parameters then they can be represented by a plane, and the equilibriums by a surface, on which there can be both fold curves and cusp points where two fold curves meet. If there are more parameters then the equilibriums form a manifold, which can have more complicated singularities; Thom called these elementary catastrophes and classified them.

He claimed that four things in particular had influenced his discoveries; first, his own previous work on singularities; second, he happened to see a sequence of models of the growing embryo; third, Sir Christopher Zeeman's use of dynamical systems to model the brain; and, fourth, his own experiments with three-dimensional light caustics. Anyone who has ever drunk a cup of white coffee in the sun will be familiar with light caustics, for the bright cusp shape on the surface of the coffee is two-dimensional caustic.

Three-dimensional caustics can be formed in mid-air by using lenses and spherical mirrors, and can be illuminated brightly by blowing smoke on them. There are three, analogous to the cusp, which are called the swallowtail and the elliptic and hyperbolic umbilics. Thom was expecting to see only the swallowtail and was astonished to discover the umbilics, which gave him the insight into the underlying mathematics. It took him about ten years, throughout the 1960s, to prove the main theorems, namely the genericity and classification of elementary catastrophes, and on the way he had to persuade colleagues to prove some of the necessary lemmas, notably Malgrange's preparation theorem and Mather's analysis of germs. In 1972 he published his remarkable book Structural stability and morphogenesis, and it is perhaps worth quoting the final sentence of that book: "At a time when so many scholars in the world are calculating, is it not desirable that some, who can, dream?"

Thom was born in Montbeliard in 1923. He studied mathematics at the Ecole Normale Superieure in Paris from 1943 until 1946, and then took a CNRS research post at Strasbourg until 1951. This was a happy period for him, for he married and began a family. He wrote his doctoral thesis in algebraic topology under Henri Cartan in 1951, and then, inspired by Charles Ehresmann, moved in a more geometric direction. After a fellowship year in Princeton from 1951 to 1952, he returned to a chair at Strasbourg.

Many mathematicians and scientists were inspired by Thom's genius, including Zeeman, who brought catastrophe theory to Britain and to the attention of the international mathematical community, and who developed many applications in the biological and behavioural sciences. One of the limitations of catastrophe theory, however, is that it is qualitative rather than quantitative (that is, invariant under smooth rather than just linear changes of co-ordinates), making it difficult sometimes, but not always, to test models numerically: indeed, there have been several notable successes.

Non-elementary catastrophe theory includes chaos theory; for example, the onset of turbulence can sometimes be modelled by a catastrophic jump from equilibrium to chaos. Chaos theory is, however, still in its infancy, with no classification theorems yet (unlike elementary catastrophe theory); nor has it shown much predictive value yet. The main contribution of chaos theory so far has been to give a better understanding of unpredictability, of how small perturbations in the initial conditions of a deterministic system can give rise to large variations in the ensuing motion. For example, building bigger computers will not necessarily lengthen accurate weather predictions.

In his later years Thom turned his attention to philosophy, and in particular to linguistics. He avoided administration and teaching, but he gave many seminars. Although his earlier theorems were profound, rigorous and beautiful, his seminars were often confusing, because his mind tended to leap ahead, leaving the audience to fill in the gaps.

But one-to-one conversations with him were marvellous: if challenged to fill a gap he would often reveal a goldmine. He showed a gentle wit, a great scepticism, and a quiet amusement at the human condition. He had original ideas about everything under the sun. His writings were often provocative in order to stimulate the reader into seeing the truth. He was awarded many honours and medals, not only in France but throughout the world.

He is survived by his wife, Suzanne, and their two daughters and son.

© The Times, 2002