# Elie Joseph Cartan

# Elie Joseph Cartan

**1869-1951**

**French Mathematician**

The career of Elie Joseph Cartan brought together four disparate mathematical fields: differential geometry, classical geometry, topology, and Lie theory. The latter was the creation of Norwegian mathematician Marius Sophus Lie (1842-1899), and concerns the application of continuous groups and symmetries in group theory. A successful and highly admired teacher, Cartan influenced a number of younger mathematicians—including his son, Henri Paul Cartan.

Born on April 9, 1869, Cartan's background was that of a peasant: his father, Joseph, was the village blacksmith in Dolomieu Isére, a town in the French Alps. Joseph and his wife Anne Cottaz Cartan had four children, of which Elie was the second. A talented student, the young Cartan attracted the attention of an inspector of primary schools, Antonin Dubost, when the latter visited Cartan's school. Dubost assisted Cartan in earning a scholarship to a lycée, or secondary school, a rare opportunity for someone of his humble origins. Cartan's advancement inspired his youngest sister Anna, who also went on to become a teacher of mathematics and author of several geometry texts.

Cartan entered the Ecole Normale Supériere in 1888, and in 1894 earned his doctorate. During this period, he first became interested in Lie's theory of continuous groups, which at that time had not attracted much attention among mathematicians. Cartan's doctoral thesis concerned the classification of semi-simple algebras, an undertaking begun by Wilhelm Killing earlier, and applied this to Lie's algebras.

His further investigations of group theory would have to wait, however, because Cartan was drafted into the French army following his graduation. He served for a year, and was discharged as a sergeant. Afterward he accepted a position as lecturer, first at the University of Montepellier, and in 1896, at the University of Lyons. Cartan remained at Lyons until 1903, during which time he continued to explore Lie's theory. It was at this point that he first began bringing it together with differential geometry, classical geometry, and topology. Also during this phase, he helped establish the foundation for the calculus of exterior differential forms, which he later applied to a variety of geometric problems.

In 1903, Cartan married Marie-Louise Bianconi, and became a professor at the University of Nancy. He remained there until 1909, at which point he moved to the Sorbonne, or the University of Paris. There he became a full professor in 1912, and he would continue at the Sorbonne for the remainder of his career.

Cartan's work continued in 1913, with the discovery of spinors, complex vectors used for developing representations of three-dimensional rotations. These in turn gained application in the development of quantum mechanics. He also continued to explore his applications of Lie theory, and its intertwining with the three other disciplines to which he had applied it.

Cartan's marriage was a happy one, and it produced four children, of whom the most famous was Henri, born in 1904. His daughter Hélène also became a mathematician. Cartan's other two children, however, met with tragedy: Jean, a composer, died of tuberculosis when he was just 25 years old, and Louis, a physicist, was executed by the Nazis in 1943 for his activities with the French Resistance.

On his 70th birthday in 1939, just before the Nazi invasion brought an end to the peaceful world he had known, Cartan was honored with a celebratory symposium at the Sorbonne. He earned a number of other honors during his career, and though he retired from the Sorbonne in 1940, in retirement he taught math at the Ecole Normale Supériere for Girls. Cartan died on May 6, 1951, in Paris.

**JUDSON KNIGHT**

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**Elie Joseph Cartan**