Élie Joseph Cartan | Biography

Élie Joseph Cartan is one of the most important mathematical figures in the first half of the 20th century. Although recognition for his many accomplishments did not come until late in his career, his intellectual influence is still felt as modern mathematics develops. From the earliest point in his career, Cartan further developed Norwegian mathematician Marius Sophus Lie's group theory which concerned continuous groupsand symmetries within them. He interworked Lie's theory and the original means in which he studied its global properties with differential geometry, classical geometry, and topology. These combined areas are still a vital part of contemporary mathematics. Cartan also formulated many original theories based on his studies. Despite making such vital contributions to mathematics, Cartan was quite modest, good humored, and easy going. He was also a gifted and well-liked teacher who could break down his intricate theories for the consumption of an average student.

Cartan was born April 9, 1869, in Dolomieu Isère, France, a village in the Alps. Of peasant descent, he was the son of the village blacksmith, Joseph, and his wife, Anne (Cottaz) Cartan. Cartan was the second oldest of four children. An inspector of primary schools, Antonin Dubost, noticed Cartan's impressive scholastic aptitude while on a visit to Cartan's school. Though most children from poor families were not put on the track to attend university, Dubost helped Cartan to win a scholarship to attend lycée (secondary school). Cartan's success as a student inspired his youngest sister, Anna, to follow in his intellectual footsteps, and become a mathematics teacher. She also published several texts on geometry.

After attending three lycées, Cartan went on to the l'Ecole Normale Supérieure in Paris in 1888, obtaining his doctorate in 1894. In his graduate thesis, Cartan began the first phase of his life's research, that of Lie's theory of continuous groups, a topic neglected by most of his contemporaries. In his thesis Cartan completed the classification of semisimple algebras begun by Wilhelm Killing during the last two decades of the 19th century, and gave a rigorous foundation for the types of Lie algebrasthat Killing had shown to be possible.

Before pursuing a career as a mathematician, Cartan was drafted into the French Army for one year after his graduation, rising to the rank of sergeant before his discharge. Cartan then served as a lecturer at two successive institutions, the University of Montpellier (1894 to 1896), and the University of Lyons (1896 to 1903). While a lecturer, he continued his mathematical studies based on Lie's theory, and began bringing together the four disparate disciplines that became the hallmark of Cartan's work: differential geometry, classical geometry, topology, and Lie theory. Cartan began by exploring the structure for associative algebras, then moved onto semisimple Lie groups and their representations. In this period (roughly from 1894 to 1904), Cartan also helped create and develop the calculus of exterior differential forms. This calculus became an important tool Cartan used in his research and he applied it to various differential geometric problems.

In 1903, Cartan's life changed in two fundamental ways. He married Marie-Louise Bianconi in that year. With her he had four children (three sons and one daughter), and enjoyed a happy home life. His eldest son, Henri Paul Cartan, born in 1904, became a prominent mathematician in his own right. His daughter Hélène also became a mathematician who taught at lycées and published some mathematical papers. Cartan's two other sons died tragic deaths. Jean, a composer, died at 25 of tuberculosis; Louis, a physicist, was arrested and imprisoned by the Nazis for his activities in the French Resistance. Louis Cartan was executed by the Nazis in 1943, but his family did not learn of his death until 1945.

In 1903, Cartan also became a professor at the University of Nancy, where he worked until 1909. Cartan moved on to the University of Paris (the Sorbonne) in 1909, where he was a lecturer from 1909 to 1912 before becoming a full professor in 1912. He remained a professor there until he retired in 1940. In his first years at the Sorbonne, in 1913, Cartan made one of his most significant contributions to math when he discovered spinors. The spinors are complex vectors used to make two-dimensional representations out of three-dimensional rotations. The spinors were important in the development of quantum mechanics.

Though Cartan continued to intertwine the four subjects mentioned earlier, at the height of his career, he continued to look at them from different perspectives. For example, after 1916, Cartan's publications focused mainly on differential geometry. Cartan developed a moving frames theory and a generalization of Daboux's kinematical theory. In 1925, Cartan refocused his attention to the study of topology. In his paper "La géométrie des espaces de Riemann" ("The Geometry of Riemann Spaces"), Cartan came up with innovative ways to study Lie groups' global properties. He demonstrated, in topological terms, that a Euclidean space and a compact group can produce a connected Lie group. And, from 1926 to 1932, Cartan published treatises concerning his geometric theory of symmetric spaces.

Cartan's continual output of important work led to his appointment as a member of the Académie Royale des Sciences in 1931, one of many honors he received late in his career. About the same time, Cartan began collaborating with his son Henri on mathematical problems, building on his son's theorems. On Cartan's 70th birthday, in 1939, the Sorbonne honored him with a celebratory symposium which praised his many mathematical accomplishments. Although Cartan retired from the Sorbonne in 1940, he remained an honorary professor there until his death. He also continued to publish mathematical treatises and his love of classroom instruction led him to teach math at the l'Ecole Normale Supérieure for Girls. Cartan died in Paris on May 6, 1951, after a long illness.