**Élie Cartan** worked on continuous groups, Lie algebras, differential equations and geometry. His work achieves a synthesis between these areas. He is one of the most important mathematicians of the first half of the 20C.

- It was Élie's exceptional abilities, together with a lot of luck, which made a high quality education possible for him.
- 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 encouraged Élie to enter the competition for state funds to allow Élie to attend a Lycée.
- Cartan became a student at the École Normale Supérieure in 1888 where he attended courses by the leading mathematicians of the day including Henri Poincaré, Charles Hermite, Jules Tannery, Gaston Darboux, Paul Appell, Émile Picard and Édouard Goursat.
- Cartan graduated in 1891 and then served for a year in the army before continuing his studies for his doctorate at the École Normale Supérieure.
- While Cartan was in the army, where he reached the rank of sergeant, his friend Arthur Tresse (1868-1958) was studying under Sophus Lie in Leipzig.
- On his return, Tresse told Cartan about Wilhelm Killing's remarkable work on the structure of finite continuous groups of transformations.
- Cartan set about completing Killing's classification and he was able to benefit greatly from a six-month visit by Sophus Lie to Paris in 1892.
- During the two years 1892-94 that Cartan spent working on his doctoral thesis, he was supported by a prestigious bursary from the Peccot Foundation.
- Cartan's 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.
- This was shown by Cartan in his thesis when he constructed each of the exceptional simple Lie algebras over the complex field.
- Cartan published full details of the classification in a third paper which was essentially his doctoral thesis.
- The eldest son, Henri Cartan, was to produce brilliant work in mathematics and has a biography in this archive.
- 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.
- In 1903 Cartan was appointed as a professor at the University of Nancy but he also taught at the Institute of Electrical Engineering and Applied Mechanics.
- Cartan worked on continuous groups, Lie algebras, differential equations and geometry.
- Joseph Wedderburn would complete Cartan's work in this area.
- 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.
- In 1899 Cartan published his first paper on the Pfaff problem Sur certaines expressions différentielles et le probleme de Pfaff Ⓣ(On some differential expressions and a problem of Pfaff).
- In this paper Cartan gave the first formal definition of a differential form.
- Cartan's papers on differential equations are in many ways his most impressive work.
- 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.
- Klein's 'Erlanger Programme' was seen to be inadequate as a general description of geometry by Weyl and Veblen, and Cartan was to play a major role.
- 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.
- 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.
- Cartan discovered the theory of spinors in 1913.
- M Cartan points out that, in their most general mathematical form, spinors were discovered by him in 1913 in his work on linear representations of simple groups, and he emphasises their connection ...
- M Cartan's book will be indispensable to mathematicians interested in the geometrical and physical aspects of group theory, giving, as it does, a complete and authoritative survey of the algebraic theory of spinors treated from a geometrical point of view.
- This was perhaps partly due to Cartan's extreme modesty.
- Just as Freud was influenced by the mechanistic world view of 19th century science, but used this background to create something new and revolutionary which has profoundly influenced 20th century thought, so Cartan built, on a foundation of the mathematics which was fashionable in the 1890's in Paris, Berlin and Göttingen, a mathematical edifice whose implications we are still investigating.
- For his outstanding contributions Cartan received many honours, but as Dieudonné explained in the above quote, these did not come until late in career.
- A celebration was held on 18 May 1939 in the Sorbonne to celebrate Cartan's 70th birthday.
- In 1969, to celebrate the 100th anniversary of Cartan's birth, a conference was held in Bucharest.
- The conference 'The Mathematical Heritage of Élie Cartan' was held in Lyon, France from 25 June to 29 June 1984 to celebrate the 115th anniversary of Cartan's birth.

Born 9 April 1869, Dolomieu (near Chambéry), Savoie, Rhône-Alpes, France. Died 6 May 1951, Paris, France.

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Group Theory, Physics

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive