Élie Cartan's mother was Anne Cottaz and his father was Joseph Cartan who was a blacksmith. The family were very poor and in late 19th century France it was not possible for children from poor families to obtain a university education. It was Élie's exceptional abilities, together with a lot of luck, which made a high quality education possible for him. When he was in primary school he showed his remarkable talents which impressed the young school inspector, later important politician, Antonin Dubost. Dubost was at this time employed as an inspector of primary schools and it was on a visit to the primary school in Dolomieu, in the French Alps, that he discovered the remarkable young Élie. Dubost was able to obtain state funds which paid for Élie to attend the Lycée in Lyons, where he completed his school education with distinction in mathematics. The state stipend was extended to allow him to study at the École Normale Supérieure in Paris.
Cartan became a student at the École Normale Supérieure in 1888 and obtained his doctorate in 1894. He was then appointed to the University at Montpellier where he lectured from 1894 to 1896. Following this he was appointed as a lecturer at the University of Lyon, where he taught from 1896 to 1903. In 1903 Cartan was appointed as a professor at the University of Nancy and he remained there until 1909 when he moved to Paris. His appointment in 1909 was as a lecturer at the Sorbonne but three years later he was appointed to the Chair of Differential and Integral Calculus in Paris. He was appointed as Professor of Rational Mechanics in 1920, and then Professor of Higher Geometry from 1924 to 1940. He retired in 1940.
He married Marie-Louise Bianconi in 1903 and they had four children, one of them Henri Cartan was to produce brilliant work in mathematics. Two other sons died tragically. Jean, a composer, died of tuberculosis at the age of 25 while their son Louis was a member of the Resistance fighting in France against the occupying German forces. After his arrest in February 1943 the family received no further news but they feared the worst. Only in May 1945 did they learn that he had been beheaded by the Nazis in December 1943. By the time they received the news of Louis' murder by the Germans, Cartan was 75 years old and it was a devastating blow for him. Their fourth child was a daughter.
Cartan worked on continuous groups, Lie algebras, differential equations and geometry. His work achieved a synthesis between these areas. He added greatly to the theory of continuous groups which had been initiated by Lie. His doctoral thesis of 1894 contains a major contribution to Lie algebras where he completed the classification of the semisimple algebras over the complex field which Killing had essentially found. However, although Killing had shown that only certain exceptional simple algebras were possible, he had not proved that in fact these algebras exist. This was shown by Cartan in his thesis when he constructed each of the exceptional simple Lie algebras over the complex field. He later classified the semisimple Lie algebras over the real field and found all the irreducible linear representations of the simple Lie algebras. He 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 applied Grassmann algebra to the theory of exterior differential forms. He developed this theory between 1894 and 1904 and applied his theory of exterior differential forms to a wide variety of problems in differential geometry, dynamics and relativity. Dieudonné writes in :-
He discussed a large number of examples, treating them in an extremely elliptic style that was made possible only by his uncanny algebraic and geometric insight and that has baffled two generations of mathematicians.
In 1945 he published the book Les systèmes différentiels extérieurs et leurs applications géométriques.
By 1904 Cartan was writing papers on differential equations and in many ways this work is his most impressive. Again his approach was totally innovative and he formulated problems so that they were invariant and did not depend on the particular variables or unknown functions. This enabled Cartan to define what the general solution of an arbitrary differential system really is but he was not only interested in the general solution for he also studied singular solutions. He did this by moving from a given system to a new associated system whose general solution gave the singular solutions to the original system. He failed to show that all singular solutions were given by his technique, however, and this was not achieved until four years after his death.
From 1916 on he published mainly on differential geometry. Klein's Erlanger Programm 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. In fact this work led Cartan to the notion of a fibre bundle although he does not give an explicit definition of the concept in his work.
Cartan further contributed to geometry with his theory of symmetric spaces which have their origins in papers he wrote in 1926. It developed ideas first studied by Clifford and Cayley and used topological methods developed by Weyl in 1925. This work was completed by 1932 and so provides :-
... one of the few instances in which the initiator of a mathematical theory was also the one who brought it to completion.
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 Élie Cartan was able to build on theorems proved by his son. Henri Cartan said :-
[My father] knew more than I did about Lie groups, and it was necessary to use this knowledge for the determination of all bounded circled domains which admit a transitive group. So we wrote an article on the subject together [Les transformations des domaines cerclés bornés, C. R. Acad. Sci. Paris 192 (1931), 709-712]. But in general my father worked in his corner, and I worked in mine.
Cartan discovered the theory of spinors in 1913. These are complex vectors that are used to transform three-dimensional rotations into two-dimensional representations and they later played a fundamental role in quantum mechanics. Cartan published the two volume work Leçons sur la théorie des spineurs in 1938.
He is certainly one of the most important mathematicians of the first half of the 20th century. Dieudonné writes in :-
Cartan's recognition as a first rate mathematician came to him only in his old age; before 1930 Poincaré and Weyl were probably the only prominent mathematicians who correctly assessed his uncommon powers and depth. This was due partly to his extreme modesty and partly to the fact that in France the main trend of mathematical research after 1900 was in the field of function theory, but chiefly to his extraordinary originality. It was only after 1930 that a younger generation started to explore the rich treasure of ideas and results that lay buried in his papers. Since then his influence has been steadily increasing, and with the exception of Poincaré and Hilbert, probably no one else has done so much to give the mathematics of our day its present shape and viewpoints.
For his outstanding contributions Cartan received many honours, but as Dieudonné explained in the above quote, these did not come until late in career. He received honorary degrees from the University of Liege in 1934, and from Harvard University in 1936. In 1947 he was awarded three honorary degrees from the Free University of Berlin, the University of Bucharest and the Catholic University of Louvain. In the following year he was awarded an honorary doctorate by the University of Pisa. He was elected a Fellow of the Royal Society of London on 1 May 1947, the Accademia dei Lincei and the Norwegian Academy. Elected to the French Academy of Sciences on 9 March 1931 he was vice-president of the Academy in 1945 and President in 1946.
Article by: J J O'Connor and E F Robertson