Subject: Rene Thom obituary
Date: Fri, 15 Nov 2002 11:52:35 -0500
From: Ron Umble
To: "'dmd1@lehigh.edu'"
Don,
People on your list-seve might be interested in this obituary that appeared
in the Times today.
Ron
-----------------------------------
November 15, 2002
Rene Thom
Creator of 'catastrophe theory' whose mathematical ventures into higher
dimensions won him the Fields Medal
René 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 Montbéliard in 1923. He studied mathematics at the Ecole
Normale Supérieure 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.
René Thom, French mathematician, Fields Medallist, philosopher, and the
creator of catastrophe theory, was born in Montbéliard on September 2, 1923.
He died at Bures-sur-Yvette on October 25, 2002, aged 79.