Eli Cardan

Cardan was born in Dolommue, Savoie, and in 1888 he became a student at the Ecole Supérieure in Paris. He laid the foundations for Lie group theory and its geometric applications. He also made significant contributions to mathematical physics, differential geometry, and group theory.

After receiving his doctorate in 1894, he taught in Montpellier and Lyon, and became a professor in Nancy in 1903. He went to Paris to teach in 1909, became a professor in 1912, and retired in 1942. He died in Paris. The mathematician Henry Gardan was his son. He once guided Chinese mathematician Chen Shengshen.

According to his own description in "Notice sur les travaux scientifiques", the subject of his work (totaling 186, published between 1893 and 1947) was the theory of Lie groups. He started with his work on basic materials on complex simple Lie algebras and organized the previous work of Christian Engel and Wilhelm Killing. This proved to be decisive, at least for classification, and he identified four main races and five special cases. He also introduced the concept of algebraic groups, which had not been seriously developed before 1950.

He also defined the general concept of antisymmetric differential forms, in the style we use today; his treatment of Lie groups through the Mayue-Cardan equation requires expression in 2-form. At that time, the concept known as Pfafian systems (i.e., systems of first-order differential equations expressed in 1-form) was very common; by introducing new variables to represent derivatives, and additional differential forms, they could express very general partial differential equation (PDE) systems. Cardan added the outer derivative as a coordinate independent operation of a completely geometric expression. This naturally led to the need to discuss the p-form for general p. Jiadan described the influence of Riquier's general PDE theory on him.

Based on these foundations-Lie groups and differential forms-he continued to complete a lot of in-depth work, as well as some common techniques, such as the moving frame method, which gradually became integrated into the mainstream of mathematics.