René Thom

1923–2002, Montbéliard, France, Bures-sur-Yvette, France

TopologyDifferential equations and dynamical systemsFields Medal 1958

French mathematician, Fields Medalist 1958, founder of cobordism theory and the Thom space, later creator of catastrophe theory on singularities of smooth maps and its applications to models of sudden change.

René Thom was born in Montbéliard, near the Swiss border, and studied at the École normale supérieure in Paris during the war. He followed Henri Cartan to Strasbourg and completed his doctorate there in 1951 under Charles Ehresmann, on fiber spaces in spheres and Steenrod squares. In 1954 he published Quelques propriétés globales des variétés différentiables, the paper that founded cobordism theory: two manifolds are cobordant if together they bound a manifold of one dimension higher, and Thom reduced the classification of manifolds in this sense to computations with Thom spaces and stable homotopy.

The work earned him the Fields Medal at Edinburgh in 1958, shared with Klaus Roth. Thom spent the early 1960s refining transversality and singularity theory — the Thom isotopy theorem and his stratification theory — and in 1964 moved to the new Institut des Hautes Études Scientifiques at Bures-sur-Yvette, where he spent the rest of his career.

From the late 1960s Thom turned to a grand program: explaining abrupt qualitative change in nature through the singularities of smooth mappings. His book Structural Stability and Morphogenesis (1972) founded what he called catastrophe theory, classifying the seven elementary catastrophes and proposing applications from embryology to linguistics. The theory was fashionable and controversial, but its mathematical core — the theory of unfoldings of singularities — became standard differential topology. He died in Bures-sur-Yvette in 2002.

Workplaces: University of Strasbourg, Institut des Hautes Études Scientifiques (IHÉS)

France

Award-winning work

Fields Medal 1958Thom received the Fields Medal for creating cobordism theory: his 1954 paper classified smooth manifolds up to cobordism by computing the cobordism ring through homotopy theory. The key device was the Thom space of a vector bundle — later fundamental in stable homotopy theory — together with the Thom isomorphism relating the cohomology of a bundle to that of its base. His methods determined completely the unoriented cobordism ring and much of the oriented one, turning classification questions about manifolds into computations in homotopy groups, and the Pontryagin–Thom construction became a permanent tool of differential topology.

Contributions, linked to the library

Library topicSmooth manifoldsCobordism theory classifies smooth manifolds by whether they jointly bound a manifold of one higher dimension, via the Thom space construction.Library topicHomology and cohomologyThe Thom isomorphism relates the cohomology of a vector bundle's Thom space to that of its base, a cornerstone of oriented cohomology theories.