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Elie Cartan worked on continuous groups, Lie algebras, differential equations and geometry. His work achieves a synthesis between these areas.
Cartan became a student at l'Ecole Normale in 1888 and obtained his doctorate in 1894. He lectured at Montpellier (1894-1896), Lyon (1896-1903), Nancy (1903-1909) and Paris (1909-1940). He had 4 children, one of them Henri Cartan was to produce brillant work in mathematics. Two others died tragically. Jean, a composer, died at the age of 25 while Louis, a physicist, was arrested by the Germans in 1942 and executed after 15 months in captivity.
Cartan added greatly to the theory of continuous groups which had been initiated by Lie. His thesis (1894) contains a major contribution to Lie algebras where he completed the classification of the semisimple algebras which Killing had essentially found. He then turned to the theory of associative algebras and investigated the structure for these algebras over the real and complex field. Wedderburn would complete Cartan's work in this area.
He then turned to representations of semisimple Lie groups. His work is a striking synthesis of Lie theory, classical geometry, differential geometry and topology which was to be found in all Cartan's work. He also applied Grassmann algebra to the theory of exterior differential forms.
By 1904 Cartan was turning to papers on differential equations and from 1916 on he published mainly on differential geometry. Klein's Erlanger Program was seen to be inadequate as a general description of geometry by Weyl and Veblen and Cartan was to play a major role. He examined a space acted on by an arbitrary Lie group of transformations, developing a theory of moving frames which generalises the kinematical theory of Darboux.
Cartan further contributed to geometry with his theory of symmetric spaces which have their origins in papers he wrote in 1926. It develops ideas first studied by Clifford and Cayley and used topological methods developed by Weyl in 1925. This work was completed by 1932.
Cartan then went on to examine problems on a topic first studied by Poincaré. By this stage his son, Henri Cartan, was making major contributions to mathematics and Elie Cartan was able to build on theorems proved by his son.
Cartan also published work on relativity and the theory of spinors. He is certainly one of the most important mathematicians of the first half of the 20 C.
References elsewhere in this archive:
Tell me about Cartan's work on quantum theory
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