René Thom
Encyclopedia
René Frédéric Thom was a French
mathematician
. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory
; he became world-famous among the wider academic community and the educated general public for one aspect of this latter interest, his work as founder of catastrophe theory
(later developed by Erik Christopher Zeeman
). He received the Fields Medal
in 1958.
, Doubs
. He was educated at the Lycée Saint-Louis
and the École Normale Supérieure
, both in Paris. He received his PhD in 1951 from the University of Paris
. His thesis, titled Espaces fibrés en sphères et carrés de Steenrod (Sphere bundles and Steenrod squares), was written under the direction of Henri Cartan
. The foundations of cobordism
theory, for which he received the Fields Medal at Edinburgh in 1958, were already present in his thesis.
After a fellowship in the United States
, he went on to teach at the Universities of Grenoble (1953–1954) and Strasbourg
(1954–1963), where he was appointed Professor in 1957. In 1964, he moved to the Institut des Hautes Études Scientifiques
, in Bures-sur-Yvette
. He was awarded the Grand Prix Scientifique de la Ville de Paris in 1974, and became a Member of the Academie des Sciences of Paris in 1976.
While René Thom is most known to the public for his development of catastrophe theory between 1968 and 1972, his earlier work was on differential topology
. In the early 1950s it concerned what are now called Thom space
s, characteristic class
es, cobordism theory, and the Thom transversality theorem. Another example of this line of work is the Thom conjecture
, versions of which have been investigated using gauge theory
. From the mid 50's he moved into singularity theory
, of which catastrophe theory is just one aspect, and in a series of deep (and at the time obscure) papers between 1960 and 1969 developed the theory of stratified sets
and stratified maps, proving a basic stratified isotopy theorem describing the local conical structure of Whitney stratified sets
, now known as the Thom-Mather isotopy theorem. Much of his work on stratified sets was developed so as to understand the notion of topologically stable maps, and to eventually prove the result that the set of topologically stable mappings between two smooth manifolds is a dense set
. Thom's lectures on the stability of differentiable mappings, given at Bonn in 1960, were written up by Harold Levine
and published in the proceedings of a year long symposium on singularities at Liverpool University during 1969-70, edited by Terry Wall. The proof of the density of topologically stable mappings was completed by John Mather in 1970, based on the ideas developed by Thom in the previous ten years. A coherent detailed account was published in 1976 by C. Gibson, K. Wirthmuller, E. Looijenga and A. du Plessis.
During the last twenty years of his life Thom's published work was mainly in philosophy and epistemology, and he undertook a reevaluation of Aristotle's writings on science.
Beyond Thom's contributions in algebraic topology, his influence on modern differential geometry, through the intensive study of generic properties
, can hardly be exaggerated.
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory
Singularity theory
-The notion of singularity:In mathematics, singularity theory is the study of the failure of manifold structure. A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width. What is meant by a singularity can be seen by dropping it on the floor...
; he became world-famous among the wider academic community and the educated general public for one aspect of this latter interest, his work as founder of catastrophe theory
Catastrophe theory
In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more general singularity theory in geometry....
(later developed by Erik Christopher Zeeman
Erik Christopher Zeeman
Sir Erik Christopher Zeeman FRS , is a Japanese-born British mathematician known for his work in geometric topology and singularity theory....
). He received the Fields Medal
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
in 1958.
Biography
René Thom was born in MontbéliardMontbéliard
Montbéliard is a city in the Doubs department in the Franche-Comté region in eastern France. It is one of the two subprefectures of the department.-History:...
, Doubs
Doubs
Doubs is a department the Franche-Comté region of eastern France named after the Doubs River.-History:As early as the 13th century, inhabitants of the northern two-thirds of Doubs spoke the Franc-Comtois language, a dialect of Langue d'Oïl. Residents of the southern third of Doubs spoke a dialect...
. He was educated at the Lycée Saint-Louis
Lycée Saint-Louis
The lycée Saint-Louis is a higher education establishment located in the VIe arrondissement of Paris, in the Latin Quarter. It is the only public French lycée exclusively dedicated to classes préparatoires aux grandes écoles...
and the École Normale Supérieure
École Normale Supérieure
The École normale supérieure is one of the most prestigious French grandes écoles...
, both in Paris. He received his PhD in 1951 from the University of Paris
University of Paris
The University of Paris was a university located in Paris, France and one of the earliest to be established in Europe. It was founded in the mid 12th century, and officially recognized as a university probably between 1160 and 1250...
. His thesis, titled Espaces fibrés en sphères et carrés de Steenrod (Sphere bundles and Steenrod squares), was written under the direction of Henri Cartan
Henri Cartan
Henri Paul Cartan was a French mathematician with substantial contributions in algebraic topology. He was the son of the French mathematician Élie Cartan.-Life:...
. The foundations of cobordism
Cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold. Two manifolds are cobordant if their disjoint union is the boundary of a manifold one dimension higher. The name comes...
theory, for which he received the Fields Medal at Edinburgh in 1958, were already present in his thesis.
After a fellowship in the United States
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
, he went on to teach at the Universities of Grenoble (1953–1954) and Strasbourg
University of Strasbourg
The University of Strasbourg in Strasbourg, Alsace, France, is the largest university in France, with about 43,000 students and over 4,000 researchers....
(1954–1963), where he was appointed Professor in 1957. In 1964, he moved to the Institut des Hautes Études Scientifiques
Institut des Hautes Études Scientifiques
The Institut des Hautes Études Scientifiques is a French institute supporting advanced research in mathematics and theoretical physics...
, in Bures-sur-Yvette
Bures-sur-Yvette
Bures-sur-Yvette is a commune in the Essonne department in Île-de-France in northern France.Inhabitants of Bures-sur-Yvette are known as Buressois.-Geography:...
. He was awarded the Grand Prix Scientifique de la Ville de Paris in 1974, and became a Member of the Academie des Sciences of Paris in 1976.
While René Thom is most known to the public for his development of catastrophe theory between 1968 and 1972, his earlier work was on differential topology
Differential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
. In the early 1950s it concerned what are now called Thom space
Thom space
In mathematics, the Thom space, Thom complex, or Pontryagin-Thom construction of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space....
s, characteristic class
Characteristic class
In mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possesses sections or not...
es, cobordism theory, and the Thom transversality theorem. Another example of this line of work is the Thom conjecture
Thom conjecture
In mathematics, a smooth algebraic curve C in the complex projective plane, of degree d, has genus given by the formulag = /2.The Thom conjecture, named after French mathematician René Thom, states that if \Sigma is any smoothly embedded connected curve representing the same class in homology as C,...
, versions of which have been investigated using gauge theory
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
. From the mid 50's he moved into singularity theory
Singularity theory
-The notion of singularity:In mathematics, singularity theory is the study of the failure of manifold structure. A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width. What is meant by a singularity can be seen by dropping it on the floor...
, of which catastrophe theory is just one aspect, and in a series of deep (and at the time obscure) papers between 1960 and 1969 developed the theory of stratified sets
Topologically stratified space
In topology, a branch of mathematics, a topologically stratified space is a space X that has been decomposed into pieces called strata; these strata are topological manifolds and are required to fit together in a certain way...
and stratified maps, proving a basic stratified isotopy theorem describing the local conical structure of Whitney stratified sets
Whitney conditions
In differential topology, a branch of mathematics, the Whitney conditions are conditions on a pair of submanifolds of a manifold introduced by Hassler Whitney in 1965...
, now known as the Thom-Mather isotopy theorem. Much of his work on stratified sets was developed so as to understand the notion of topologically stable maps, and to eventually prove the result that the set of topologically stable mappings between two smooth manifolds is a dense set
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
. Thom's lectures on the stability of differentiable mappings, given at Bonn in 1960, were written up by Harold Levine
Harold Levine
Harold Levine is an American mathematician who is professor emeritus at Stanford university. He specializes in wave motion and optics. In 1954, he was awarded a Guggenheim fellowship.-External links:...
and published in the proceedings of a year long symposium on singularities at Liverpool University during 1969-70, edited by Terry Wall. The proof of the density of topologically stable mappings was completed by John Mather in 1970, based on the ideas developed by Thom in the previous ten years. A coherent detailed account was published in 1976 by C. Gibson, K. Wirthmuller, E. Looijenga and A. du Plessis.
During the last twenty years of his life Thom's published work was mainly in philosophy and epistemology, and he undertook a reevaluation of Aristotle's writings on science.
Beyond Thom's contributions in algebraic topology, his influence on modern differential geometry, through the intensive study of generic properties
Generic property
In mathematics, properties that hold for "typical" examples are called generic properties. For instance, a generic property of a class of functions is one that is true of "almost all" of those functions, as in the statements, "A generic polynomial does not have a root at zero," or "A generic...
, can hardly be exaggerated.
See also
- Dold–Thom theoremDold–Thom theoremIn algebraic topology, the Dold–Thom theorem, proved by , states that the homotopy group πi of the infinite symmetric product SP of X is the i-th singular reduced homology group of "X", usually denoted by Hi with an added ~ on the H....
- Quelques propriétés globales des variétés differentiables
- Thom isomorphism
- Pontryagin-Thom construction
External links
- Washington Post Online edition (free registration)
- Meeting René THOM