Élie Cartan

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Élie Cartan

Élie Joseph Cartan (born April 9, 1869 in Dolomieu , Dauphiné ; † May 6, 1951 in Paris ) was a French mathematician who made significant contributions to the theory of Lie groups and their applications. He also made significant contributions to mathematical physics and differential geometry .

Life

Cartan's father was a blacksmith and the family would not have been able to finance his higher education if his talent had not been noticed by a school inspector while attending the elementary school in Dolomieu. He received a scholarship to attend high school (Lycée) in Lyon and then from 1888 the elite school École normal supérieure in Paris . After his doctorate in 1894, he taught at the University of Montpellier and from 1896 to 1903 at the University of Lyon. In 1903 he became a professor in Nancy . In 1909 he finally began to teach in Paris, where he was a lecturer at the Sorbonne and in 1912 he was appointed professor of analysis. In 1920 he became professor of rational mechanics and in 1924 of geometry. During the First World War he worked in the hospital of the Ecole Normale Supérieure, but was still active in research. In 1940 he retired.

He had been married to Marie-Luise Bianconi since 1903, with whom he had four children. His son Henri Cartan also became an eminent mathematician. Élie Cartan's sister Anna (1878-1923) studied at the École normal de jeune filles in Sèvres and in 1904 received her Agrégation in mathematics. She then taught in the preparatory service of math teachers at her alma mater. His daughter Hélène (1917–1952) was a mathematics teacher (she studied at the Ècole normal supérieure in Paris), also published in the Comptes-rendus (1942), but soon fell ill with tuberculosis.

In 1922–1932, Cartan corresponded with Albert Einstein on the theory of distant parallelism based on the torsion discovered by Cartan . The Einstein-Cartan theory (ECT) is a synthesis of this theory with Einstein's General Theory of Relativity (ART).

In 1915 Cartan was president of the Société Mathématique de France . In 1931 he became a member of the Académie des Sciences . In 1949 he was elected to the National Academy of Sciences . A mathematics prize named after him ( Prix ​​Élie Cartan ) is awarded by the Académie des sciences. The lunar crater Cartan and the asteroid (17917) Cartan are named after him.

plant

Élie Cartan is best known for his studies on the classification of semi-simple complex Lie algebras and his contributions to differential geometry . Many concepts of the theory of Lie algebras such as Cartan subalgebras , the Cartan involution , the Cartan criterion and the Cartan matrix are named after him. In differential geometry, the Cartan derivative and Maurer-Cartan equations bear his name; sometimes connections on principal bundles (main fiber bundles ) are also referred to as Cartan connections .

He showed that even in Newtonian physics, due to the equivalence principle, force-free movements can be interpreted as straight movements along a geodesic in a curved " Newton-Cartan space-time " (similar to Einstein's theory of gravity , but with an absolute time in the Newtonian sense).

According to his own admission in his work Notice sur les travaux scientifiques , his main contribution to mathematics was the further development of the theory of Lie groups and Lie algebras (first in his dissertation in 1894). In continuation of the work of Wilhelm Killing and Friedrich Engel , he worked on complex simple Lie algebras. Here he identified the 4 main families and the 5 exceptional cases, with which a complete classification was achieved. He also introduced the concept of the algebraic group , but it did not experience serious development until after 1950.

He defined the uniform notation of alternating differential forms as it is still used today. His approach to the Lie groups using the Maurer-Cartan equations required second order equations. At that time only first order equations ( Pfaff's forms ) were used. With the introduction of the 2nd order for derivatives and other orders, the formulation of comparatively general systems of partial differential equations became possible. Cartan introduced outer derivation as a completely geometric and coordinate-independent operation. This naturally leads to the need to investigate differential forms of any degree p. As Cartan reports, he was influenced by the general theory of partial differential equations as described by Riquier.

Cartan discovered the Spinor concept in 1913 in an essay on representation theory by Liegruppen , but it only attracted greater attention after the Dirac equation was discovered in 1928, and the name Spinor was coined in 1929 by the physicist Paul Ehrenfest . Cartan came back to spinors extensively in his 1938 Lectures on Spinors.

With these foundations - Lie groups and higher order differential equations - he created a comprehensive work and introduced some basic techniques such as moving frames , which later integrated into the mainstream of mathematical methods.

In the Notice sur les Travaux Scientifiques he divides his work into fifteen parts. In modern terminology, these are:

  • Lie groups
  • Representations of Lie groups
  • Hypercomplex numbers , division algebra
  • Partial differential equations, Cartan-Kähler theorem
  • Equivalence theory
  • Integrable systems, theory of prolongations and involution systems
  • Infinite dimensional groups and pseudo groups
  • Differential geometry and accompanying multi-legs (moving frames, repere mobile)
  • General spaces with structural group and connections, Cartan connection, holonomy, Weyl tensor
  • Geometry and topology of Lie groups
  • Riemannian geometry
  • Symmetrical spaces
  • Topology of compact groups and their homogeneous spaces
  • Integral invariants and classical mechanics
  • General relativity and spinors

He was a pioneer in many of these areas. Most - but not all - topics on which he was the first to advance relatively isolated and not understood by his contemporaries, were taken up and developed by later mathematicians.

Cartan gave several plenary lectures at the International Congress of Mathematicians : in Oslo 1936 (Quelques aperçus sur le rôle de la théorie des groupes de Sophus Lie dans le développement de la géométrie moderne), Toronto 1924 (La théorie des groupes et les recherches récentes de géométrie différentielle) and Zurich 1932 (Sur les espaces riemanniens symétriques).

Fonts

  • Oeuvres complètes, 3 parts in 6 volumes, Paris 1952 to 1955, reprint Edition du CNRS 1984:
    • Part 1: Groupes de Lie. (In 2 volumes), 1952.
    • Part 2, Volume 1: Algèbre, formes différentielles, systèmes différentiels. 1953.
    • Part 2, Volume 2: Groupes finis, Systèmes différentiels, théories d´équivalence. 1953.
    • Part 3, Volume 1: Divers, géométrie différentielle. 1955.
    • Part 3, Volume 2: Géométrie différentielle. 1955.
  • Geometry of Riemannian Spaces. Brookline, Massachusetts, 1983, first La geometrie des espaces de Riemann. Gauthiers-Villars, 1925.
  • On manifolds with affine connection and the general theory of relativity. Naples, Bibliopolis 1986.
  • The Theory of Spinors. Paris, Hermann 1966 (first as Lecons sur le theorie des spineurs, Hermann 1938).
  • Lecons on the theory of space a connexion projective. Gauthiers-Villars, 1937.
  • The parallelism absolu et the theory unitaire du champ. Hermann, 1932.
  • La theory des groupes finis et continus et l'analysis situs. Gauthiers-Villars, 1930.
  • Lecons sur la geometry projective complexes. Gauthiers-Villars, 1931.
  • Lecons on the geometry of the space de Riemann. Gauthiers-Villars, 1928.
  • Lecons sur les invariants integraux. Hermann, Paris, 1922.
  • Notice sur les travaux scientifiques , Gauthier-Villars 1974

literature

See also

Web links

Individual evidence

  1. ^ Yvette Kosmann-Schwarzbach: Women mathematicians in France in the mid-twentieth century (Arxiv 2015)
  2. ↑ R. Debever: Albert Einstein - Elie Cartan. Letters on absolute parallelism 1929-1932. Princeton University Press.
personal data
SURNAME Cartan, Elie
ALTERNATIVE NAMES Cartan, Élie Joseph
BRIEF DESCRIPTION French mathematician who contributed to the theory of Lie groups and their applications
DATE OF BIRTH April 9, 1869
PLACE OF BIRTH Dolomieu , Dauphiné
DATE OF DEATH May 6, 1951
Place of death Paris