Élie Cartan

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

Life

Cartan's father was a blacksmith and the family could not finance his higher education when his talent would not have noticed a school inspector attended the primary school in Dolomieu. He received a scholarship to attend the High School ( Lycée ) in Lyon and then in 1888 the elite École Normale Supérieure in Paris. After receiving 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 received a professorship in Analysis. In 1920 he was appointed professor of rational mechanics and 1924 for geometry. During World War II he worked at the Hospital of the Ecole Normale Supérieure, but remained scientifically active. In 1940 he retired.

He was married in 1903 to Marie- Louise Bianconi, with whom he had four children. His son Henri Cartan was also an important mathematician.

A named after him mathematics Price ( Prix Élie Cartan ) is awarded by the Academie des Sciences.

Work

Élie Cartan is primarily known for his studies on the classification of semisimple complex Lie algebras and his contributions to differential geometry. According to him, many concepts of the theory of Lie algebras such Cartan subalgebras, the Cartan involution and the Cartan matrix are named. In the differential geometry of the Cartan - Maurer - Cartan derivative and equations bear his name; sometimes relationships on principal bundles ( principal bundle ) are called Cartan connections.

By his own admission in his work Notice sur les travaux scientifiques his main contribution to mathematics was the development of the theory of Lie groups and Lie algebras (first in his dissertation, 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 five exceptional cases, whereby a complete classification has been achieved. He also introduced the concept of algebraic group, but the learned only after 1950 serious development.

He defined the uniform notation of alternating differential forms, as they are still used today. His approach to Lie groups by using the Maurer - Cartan equations required equations of 2nd order. At that time, only 1st-order equations ( Pfaffian form) were used. With the introduction of the second -order derivatives and further orders for the formulation comparatively general systems of partial differential equations was possible. Cartan introduced the exterior derivative one as a completely geometric and coordinate- independent operation. This naturally leads to the need to investigate differential forms of arbitrary degree p. How Cartan reports that he has been influenced by the general theory of partial differential equations, as described by Riquier.

With these basics - Lie groups and differential equations of higher order - he created a comprehensive work and led some basic techniques such as the frame fields (moving frames ), which later became integrated into the mainstream of mathematical methods.

Add to Travaux he divided his work into fifteen sections. In modern terminology, these are:

  • Lie groups
  • Representations of Lie groups
  • Hyper complex numbers, division algebra
  • Partial differential equations, Cartan - Kähler theorem
  • Equivalence theory
  • Integrable systems, theory of prolongations and Involutionssysteme
  • Infinite-dimensional groups and pseudo- groups
  • Differential Geometry and accompanying Much legs (moving frames, repere mobile)
  • Public spaces with structure group and contexts, Cartan connection, holonomy, Weyl tensor
  • Geometry and Topology of Lie groups
  • Riemannian geometry
  • Symmetric spaces
  • Topology of compact groups and their homogeneous spaces
  • Integral invariants and classical mechanics
  • General Relativity and spinors

In many of these areas he was a pioneer. Most - but not all - issues on which it is relatively isolated and progressed by contemporaries misunderstood as the first, have been taken up and expanded by later mathematicians.

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

Writings

  • Oeuvres complètes, 3 parts in 6 volumes, Paris 1952-1955, Reprint Edition du CNRS 1984: Part 1: Groupes de Lie. (In 2 volumes ), 1952.
  • Part 2, Volume 1: Algèbre, formes differential, systèmes différentiels. In 1953.
  • Part 2, Volume 2: Groupes finis, Systèmes différentiels, théories d' équivalence. In 1953.
  • Part 3, Volume 1: Divers, géométrie différentielle. In 1955.
  • Part 3, Volume 2: Geometry tools différentielle. In 1955.
France French Academy of Sciences Semisimple module Friedrich Engel (mathematician) Differentialform Hypercomplex number Integrated Authority File
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