Cartan, Élie Joseph (ālēˈ zhôzĕfˈ kärtäNˈ) [key], 1869–1951, French mathematician. The son of a village blacksmith, he graduated from the École normale and taught at the universities of Montpellier, Lyons, Nancy, and finally Paris, where he was professor from 1912 to 1940. He developed powerful methods of attacking problems in fields related to modern topology, notably Lie groups, differential systems, and differential geometry; his discoveries are basic to mathematical formulations of quantum mechanics and general relativity. New applications are still found for his work, which is collected in *Oeuvres complètes* (1952–55). The importance of his contributions was recognized belatedly with his election to the French Academy of Sciences in 1931. His son, **Henri Cartan** äNrēˈ, 1904–2008, was also a mathematician, and was one of the founding members of the Bourbaki group (see Bourbaki, Nicolas), which sought to establish a rigorous foundation for modern mathematics. Cartan was noted especially for his work in homological algebra (the application of algebra to topology). He taught at the Univ. of Strasbourg, École Normale Supériere, and Univ. of Paris-Sud.

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