Élie Cartan

Élie Joseph Cartan was a French mathematician of immense influence. He is widely regarded as one of the greatest mathematicians of the twentieth century. His work helped build the foundation for modern mathematics. He focused on the study of shapes, movement, and change. This field is often called the development of analysis on differentiable manifolds. This subject connects Lie groups, differential geometry, and partial differential systems into a unified tool.

Cartan was born on April 9, 1869, in Dolomieu, France. He grew up in a working-class home. His father, Joseph, was a village blacksmith. Élie recalled hearing the blows of the anvil at dawn. His mother, Anne, worked with a spinning-wheel. He was a very bright but shy student. At age ten, he passed a contest for a scholarship. He later studied at the École Normale Supérieure. There, he learned from famous thinkers like Henri Poincaré.

One of Cartan's primary focuses was the theory of Lie groups. A Lie group is a set of transformations that describe how objects move or change. Earlier, Wilhelm Killing had begun classifying simple complex Lie algebras. However, Killing's proofs were often defective. Cartan provided a rigorous foundation for this local theory. He proved the existence of exceptional Lie algebras for each type. He also solved the problem of classifying simple real Lie algebras. To do this, he introduced the concept of the weight of a representation.

Cartan's work on Lie groups led to unexpected discoveries in physics. While studying the linear representations of orthogonal groups, he discovered spinors in 1913. Spinors are mathematical objects that later became essential to quantum mechanics. Later in his career, Cartan turned to topological questions. He studied the global properties of compact groups. He showed that a connected Lie group is a product of a Euclidean space and a compact group. He also found ways to determine Betti numbers using algebraic questions.

Cartan also explored infinite-dimensional versions of these groups. He called these Lie pseudogroups. A pseudogroup is a set of transformations between subsets of a space. It is similar to a group, but the composition of two transformations is not always possible. Cartan studied primitive pseudogroups of complex analytic transformations. He proved that these belong to one of six specific classes. These classes include transformations that preserve volumes or multiply volumes by a constant Jacobian.

Perhaps his most profound achievement was in the theory of differential systems. These systems are sets of equations used to describe change. Most mathematicians used specific variables to solve these problems. Cartan broke with this tradition. He sought to solve problems in a completely invariant fashion. This is known as a coordinate-free geometric formulation. He used a tool called exterior differential forms to achieve this. This method allowed him to define a "general" solution for any arbitrary system.

To find all possible solutions, Cartan used a method called prolongation. This involves adding new unknowns and new equations to a system. This process makes a singular solution become a general solution of the new system. While Cartan showed this worked for every example he treated, he did not prove it worked for every possible system. A mathematician named Masatake Kuranishi eventually provided that proof in 1955. Cartan's work remains central to how we understand complex mathematical structures today.

Cartan was also a dedicated educator and a father to mathematicians. His son, Henri Cartan, became an influential mathematician in algebraic topology. One of his students, Shiing-Shen Chern, noted that Cartan's mind never rested. Cartan could recall complex papers on Lie algebras by heart. He would often send students letters with new questions after a meeting. He served as a professor at the Sorbonne starting in 1912. He died in Paris in 1951. In 1976, a lunar crater was named in his honor.