René Thom

René Thom ( born September 2, 1923 in Montbéliard, † October 25, 2002 in Bures -sur -Yvette ) was a French mathematician and philosopher, who in 1958 was awarded for outstanding contribution to mathematics with the Fields Medal.

Life

Thom, whose parents were shopkeepers, attended primary school from 1931, Montbeliard, then the local Collège Cuvier, he received his undergraduate degree ( Bachelor ) in elementary mathematics in 1940 from Besançon. Further training was provisionally interrupted by the Second World War. His parents sent him to safety to his brother in the south and the two made ​​their way to Switzerland. In 1941 he returned to France, in Lyon took his studies and received in the same year his bachelor's degree in philosophy. He then returned to his parents shortly thereafter to study in Paris.

First Thom attended the Lycée Saint -Louis in Paris, to compete at the École Normale Supérieure, but this came about after the next start of 1943. The states at the school and in Paris were difficult as Paris was now occupied by the German Wehrmacht. At the ENS Thom was mainly influenced by Henri Cartan. In 1946 he finished the École Normale Supérieure and ( Henri Cartan following) went to Strasbourg to accept a research position, where he also received his doctorate in 1951 at Cartan (Fibre spaces into spheres and Steenrod squares ). In his dissertation he considered vector bundles over manifolds and proved the now called the Thom isomorphism isomorphism between the cohomology of the base and the cohomology of the Thom space now called the one-point compactification of the total space. The isomorphism is realized by the cup-product with the Thom class. Thom showed further that let the boots -Whitney classes of the vector bundle calculated by application of the Steenrod operations on the Thom class. In Strasbourg, he was also influenced by Charles Ehresmann, Georges Reeb, Wu Wenjun and Jean -Louis Koszul. In the same year he traveled to the USA where he met in Princeton with Albert Einstein, Hermann Weyl ( he attended the last lectures) and Norman Steenrod and was able to attend the seminars by Kunihiko Kodaira and Eugenio Calabi.

Thom returned to France and taught from 1953 to 1954 in Grenoble, then as a professor (successor of Claude Chabauty ) in Strasbourg from 1954 to 1963 In 1958 he was awarded the Fields Medal ( plenary lecture at the International Congress of Mathematicians in Edinburgh. Des Variétés triangulées aux variétés différentiables ). In his 1954 paper published in " Quelques propriétés the global variétés differentiables " he devoted himself to going back to Norman Steenrod problem, let homology classes represent in themselves as images of the fundamental class of a manifold. He proved that this problem is dual to the question of which Kohomologieklassen represent in itself as a pullback of the Thom class of the universal bundle. (This is an environment of an embedding in a. ) To this dual problem he could specify a number of positive and negative examples using Steenrod operations. Another result of this work was an isomorphism between the abstract and the stable homotopy groups defined Kobordismusgruppen the Thom space of the universal vector bundle. This allowed Thom, the Kobordismusgruppen to calculate (up to its torsional component), for which he was awarded the Fields Medal in 1958. An immediate corollary to calculate the Kobordismusgruppen was proven by Friedrich Hirzebruch signature theorem, and the first proof of the Atiyah-Singer index theorem used Thoms Kobordismustheorie. In connection with his work differential- Thom proved many basic sets, including the eponymous Transversalitätssatz.

1956/57, and 1961 he was a visiting scientist at the Institute for Advanced Study. In 1964 he went to the Institut des Hautes Études Scientifiques in Bures -sur- Yvette, where at the time Alexander Grothendieck worked. According to Thom whose mathematical achievements in him completely alien abstract algebraic alignment meant that he felt pushed to the edge and he began to deal with biology and philosophy. In this context, he developed the theory of catastrophism, which he published in a book in 1972. A major goal was the mathematical understanding of the biological pattern formation ( morphogenesis ), also the title of the book, in which he published the theory. In 1974, the Grand Prix Scientifique de la Ville de Paris he was awarded in 1990 he became an honorary member of the London Mathematical Society ( London Mathematical Society ).

Main area of his mathematical work was the differential topology, where Thom provided numerous fundamental contributions. Known to a wider audience but he was by his catastrophe theory, an early and important contribution to the field of chaos research.

In differential topology Thom is the creator of Kobordismentheorie. Two manifolds are kobordant if their union complete edge is a third variety. The spherical surface, for example, is neutral kobordant, since it is the edge of the solid sphere. The idea came from Pontryagin, but has been reduced by Thom on the homotopy theory, which allowed the calculation of the Kobordismusgruppen. Especially for this result, he received the Fields Medal in 1958.

In 1970 he received the first Brouwer Medal. In 1970 he was invited speaker at the International Congress of Mathematicians in Nice ( Structure locale of morphismes analytiques ) and 1962 in Stockholm ( Equivalence of applications topologique polynomial ).

René Thom was married and had three children.