René Thom

René Thom in Nice , 1970

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

The main area of ​​his mathematical work was differential topology , where Thom made numerous fundamental contributions. Thom is the creator of the cobordism theory . Two manifolds are co-ordinate if their union is the complete boundary of a third manifold. The spherical surface z. B. is zero-kobordant because it is the edge of the full sphere. The idea comes from Pontryagin , but was traced back by Thom to the homotopy theory, which made it possible to calculate the cobordism groups. For this result in particular, he received the Fields Medal in 1958.

Life

Thom, whose parents were shop owners, attended elementary school in Montbéliard from 1931, then the local Collège Cuvier, he received his intermediate diploma (bachelor's degree) in elementary mathematics in 1940 from Besançon . Further training was temporarily interrupted by the Second World War. His parents sent him to his brother in the south to be on the safe side, and the two made their way to Switzerland . In 1941 he returned to France, resumed his studies in Lyon and received his intermediate diploma in philosophy in the same year. He then returned to his parents and shortly thereafter studied in Paris .

First, Thom lived in the Lycée Saint-Louis near Paris to apply to the École normal supérieure , but this only came about the next time in 1943. Conditions at the school and in Paris were difficult because Paris was meanwhile occupied by the German Wehrmacht. At the ENS Thom was mainly influenced by Henri Cartan . In 1946 he finished the École Normale Supérieure and went (following Henri Cartan) to Strasbourg to accept a research position, where he also received his doctorate in 1951 at Cartan ( Fiber Spaces in Spheres and Steenrod Squares ). In his dissertation he looked vector bundles over manifolds and proved the day as Thom isomorphism isomorphism designated between the cohomology of the base and the cohomology of today as Thom space designated Einpunktkompaktifizierung of the total space. The isomorphism is realized by the cup product with the Thom class . Thom also showed that the Stiefel-Whitney classes of the vector bundle can be calculated by applying the Steenrod operations to 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 Albert Einstein , Hermann Weyl (he attended his last lectures) and Norman Steenrod in Princeton and attended the seminars of Kunihiko Kodaira and Eugenio Calabi .

René Thom (left) with Jean-Pierre Serre (3rd from left) and others in Oberwolfach in 1949

Thom returned to France and taught in Grenoble from 1953 to 1954 , then as professor (successor to Claude Chabauty ) in Strasbourg from 1954 to 1963. In 1958 he received the Fields Medal (plenary lecture at the International Congress of Mathematicians in Edinburgh : Des Variétés triangulées aux variétés différentiables ). In his work "Quelques propriétés globales des variétés differentiables", published in 1954, he devoted himself to Norman Steenrod 's problem of which homology classes can be represented as images of the fundamental class of a manifold . He proved that this problem is dual to the question of which cohomology classes can be represented as a pullback of the Thom class of the universal bundle . (Here an environment is an embedding of in one .) With the help of Steenrod operations he was able to give numerous positive and negative examples of this dual problem. Another result of the work was an isomorphism between the abstractly defined cobordism groups and the stable homotopy groups of the Thom space of the universal vector bundle. This made it possible for Thom to calculate the cobordism groups (except for their torsion component), for which he received the Fields Medal in 1958. A direct corollary for the calculation of the cobordism groups was the signature theorem proved by Friedrich Hirzebruch , also the first proof of the Atiyah-Singer index theorem used by Thom's cobordism theory. In connection with his differential topological work, Thom proved numerous basic theorems, including the transversality theorem named after him . ${\displaystyle H_{*}(X)}$ ${\displaystyle H^{*}(N/\partial N)}$ ${\displaystyle N}$ ${\displaystyle X}$ ${\displaystyle R^{n}}$

In 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 Alexander Grothendieck was also working at the time . According to Thom, his mathematical successes in the completely foreign abstract algebraic orientation led him to feel marginalized and he began to deal with biology and philosophy. In this context he developed the catastrophe theory , which he published in a book in 1972. A main goal was the mathematical understanding of the biological formation ( morphogenesis ), including the title of the book in which he published the theory. The book was published in 1972 but had been in the works since the mid-1960s, and Thom often lectured on its contents before the book was published. The catastrophe theory enabled qualitative investigations of dynamic systems even in sciences that were still little mathematically developed at that time and found a great response in the media, especially after further development by mathematicians such as Erik Christopher Zeeman . In 1974 he was awarded the Grand Prix Scientifique de la Ville de Paris and in 1990 he became an honorary member of the London Mathematical Society .

In 1970 he received the first Brouwer medal . In 1970 he was invited speaker at the International Congress of Mathematicians in Nice ( Structure locale des morphismes analytiques ) and in 1962 in Stockholm ( Equivalence topologique des applications polynomiales ). In 1975 Thom was elected to the American Academy of Arts and Sciences , and in 1978 a member of the Leopoldina .

René Thom was married and had three children.

See also

• Thom room
• Thom conjecture

Fonts (selection)

• Oeuvres mathématiques, Société Mathématique de France, Volume 1, 2017, Volume 2, 2019

Differential topology

• "Espaces fibers en sphères et carrés de Steenrod" (PDF; 7.6 MB), Annales Scientifiques de l'École Normale Supérieure (3) 69, (1952), 109-182 (Thom isomorphism, Stiefel-Whitney classes)
• "Quelques proprietes globales des varietes differentiables" , Comm.Math.Helvetici 28 (1954), pp. 17-86, (Steenrod problem and calculation of the cobordism groups)
• "Un lemme sur les applications différentiables", Bol. Soc. Mat. Mexicana (2) 1 (1956), pp. 59-71 (proof of the transversality theorem)
• (with Albrecht Dold ) "Quasi-Fibers and Infinite Symmetrical Products" , Ann. of Math. (2) 67 (1958), pp. 239-281
• "Des variétés triangulées aux variétés différentiables" (PDF; 1.0 MB), Proc. Boarding school Congress Math. 1958, pp. 248-255, Cambridge Univ. Press, New York

Disaster theory

• Stabilité structurelle et morphogénèse - essai d'une théorie générale des modèles . Benjamin, Reading / Massachusetts 1972, 2nd edition InterÉditions, Paris 1977, engl. Structural Stability and Morphogenesis - an outline of a general theory of models . Addison-Wesley, 2nd expanded edition 1989, ISBN 0-201-09419-3
• Esquisse d'une semiophysique . Paris: InterEditions, 1988, ISBN 978-2-7296-0131-7
• Apologie du logos . Paris: Hachette, 1990, ISBN 978-2-01-014836-1
• Prédire n'est pas expliquer . (Conversations with Emile Noël), Paris 1999 (TB), ISBN 978-2-08-081288-9

Autobiography

• in Atiyah, Iagolnitzer (editor) Fields Medaillists Lectures , World Scientific 1997, pp. 71ff

literature

• Heinz Hopf , The work of René Thom , laudation for the Fields Medal for Thom, Proc. Int. Congress Mathem., Edinburgh 1958
• Athanase Papadopoulos (ed.) René Thom: Portrait mathématique et philosophique. CNRS Editions, Paris, 2018, 460 p. ISBN 978-2-271-11827-1
• Special issue on Thom in Publ Math. IHES, Volume 68, 1988, numdam , therein:
  • André Haefliger Un aperçu de l'oeuvre de Thom en topologie différentielle (jusqu'en 1957), pp. 13-18.
  • Bernard Teissier Travaux de Thom sur les singularités, pp. 19-25.
  • Thom's list of publications, pp. 9-11
• Christopher Zeeman Controversy in science: on the ideas of Daniel Bernoulli and René Thom , Nieuw Archief voor Wiskunde, Volume 11, 1993, Issue 3, pp. 257-282.
• On Thom in Bulletin of the American Mathematical Society, Volume 41, 2004, Issue 3, Online :
  • Michael Atiyah The impact of Thom's Corbordism Theory , pp. 337-340
  • Jean-Pierre Bourguignon René Thom: "Mathématicien et apprenti philosophe" , pp. 273/274
  • Dennis Sullivan Thom's work on geometric holonomy and bordism , pp. 341-450

Specifically on Thom's catastrophe theory in psychoanalysis:

• Michèle Porte La Dynamique Qualitative en Psychanalyse , 1994

Web links

• John J. O'Connor, Edmund F. Robertson :  René Thom. In: MacTutor History of Mathematics archive .
• René Thom in the Mathematics Genealogy Project (English)

Individual evidence

1. ↑ Thom's autobiography in Atiyah, Iagolnitzer Fields Medaillists Lectures 1997
2. ↑ According to Mctutor (see web links) he was appointed professor in 1957, according to his autobiography as early as 1953 and he only changed between Grenoble and Strasbourg in the first few months.
3. ↑ Autobiography in Atiyah, Iagolnitzer. Relations with my colleague Grothendieck were less agreeable to me. His technical superiority was crushing. His seminar attracted the whole of Paris mathematics, whereas I had nothing new to offer.