Élie Cartan facts for kids
Quick facts for kids
Élie Cartan
|
|
|---|---|
Professor Élie Joseph Cartan
|
|
| Born | 9 April 1869 Dolomieu, Isère, France
|
| Died | 6 May 1951 (aged 82) Paris, France
|
| Nationality | French |
| Alma mater | University of Paris |
| Known for | Lie groups (Cartan's theorem) Vector spaces and exterior algebra Differential geometry Special and general relativity Differential forms Quantum mechanics (spinors, rotating vectors) List of things named after Élie Cartan |
| Awards | Leconte Prize (1930) Lobachevsky Prize (1937) President of the French Academy of Sciences (1946) Fellow of the Royal Society (1947) |
| Scientific career | |
| Fields | Mathematics and physics |
| Institutions | University of Paris École Normale Supérieure |
| Thesis | Sur la structure des groupes de transformations finis et continus (1894) |
| Doctoral advisor | Gaston Darboux Sophus Lie |
| Doctoral students | Charles Ehresmann Mohsen Hashtroodi Kentaro Yano |
| Other notable students | Shiing-Shen Chern |
Élie Joseph Cartan (born April 9, 1869 – died May 6, 1951) was a very important French mathematician. He did groundbreaking work on Lie groups, which are special types of mathematical groups. He also worked on differential geometry, a field that studies shapes and spaces using calculus.
Cartan also made big contributions to general relativity, which is Albert Einstein's theory about gravity. His work also helped in the development of quantum mechanics, the science of how tiny particles behave. Many people consider him one of the greatest mathematicians of the 20th century.
His son, Henri Cartan, also became a famous mathematician.
Contents
Élie Cartan's Early Life and Education
Élie Cartan was born on April 9, 1869, in a small village called Dolomieu, Isère, in France. His father, Joseph Cartan, was the village blacksmith. Élie remembered hearing the sound of the anvil every morning. His mother, Anne Cottaz, was very busy taking care of the children and the house.
Élie had an older sister, Jeanne-Marie, who became a dressmaker. His younger brother, Léon, followed in their father's footsteps and became a blacksmith. His younger sister, Anna Cartan, became a mathematics teacher, partly inspired by Élie.
How Élie Started His Studies
Élie went to elementary school in Dolomieu and was the best student there. One of his teachers, M. Dupuis, noticed how smart he was. Élie was shy but had a bright mind and an excellent memory.
A local representative, Antonin Dubost, visited the school and was very impressed by Élie's skills. He suggested that Élie try for a scholarship to a lycée (a type of high school). Élie prepared with his teacher and passed the test when he was just ten years old.
He spent five years at the College of Vienna and then two years at the Lycée of Grenoble. In 1887, he moved to Paris to study sciences at the Lycée Janson-de-Sailly. There, he became friends with Jean Baptiste Perrin, who later became a famous physicist.
University Life and Early Career
In 1888, Élie Cartan joined the École Normale Supérieure, a very prestigious school in France. He learned from many famous mathematicians like Henri Poincaré, whose lectures he admired the most.
After graduating in 1891, Cartan served one year in the French army as a sergeant. Then, he returned to the École Normale Supérieure. He started working on classifying simple Lie groups, a topic that was first explored by Wilhelm Killing. In 1892, Sophus Lie, a key figure in this field, met Cartan in Paris for the first time.
In 1894, Cartan completed his important doctoral paper, called The structure of finite continuous groups of transformations. After that, he worked as a lecturer at the University of Montpellier and then at the University of Lyon.
Family Life and Professorship
In 1903, while in Lyon, Cartan married Marie-Louise Bianconi. That same year, he became a professor at the University of Nancy. Their first son, Henri Cartan, was born in 1904 and later became a famous mathematician. Their second son, Jean Cartan, born in 1906, became a composer.
In 1909, Cartan moved his family to Paris and became a lecturer at the Sorbonne, part of the University of Paris. In 1912, he became a full professor there, thanks to a recommendation from Poincaré. He taught at the Sorbonne until he retired in 1940. He spent his last years teaching mathematics at the École Normale Supérieure for girls.
Cartan was recognized internationally for his work. In 1921, he became a foreign member of the Polish Academy of Learning. In 1937, he joined the Royal Netherlands Academy of Arts and Sciences. He also helped organize international science conferences.
Élie Cartan passed away in Paris in 1951 after a long illness. In 1976, a crater on the Moon was named after him to honor his contributions.
Élie Cartan's Mathematical Discoveries
Cartan's work focused on a field called "analysis on differentiable manifolds." This is a central part of modern mathematics. He helped shape and advance this area, especially with Lie groups, partial differential systems, and differential geometry. These areas are now connected and are powerful tools in mathematics.
Understanding Lie Groups
For about 30 years after his doctoral paper, Cartan was almost alone in studying Lie groups. These groups describe transformations that change smoothly, like rotations.
In 1888, Wilhelm Killing started studying these groups more deeply. At first, mathematicians only looked at local properties, meaning what happens very close to a specific point. Cartan's main goal in his thesis was to make Killing's work more precise. He also proved that certain special Lie algebras (which describe the local properties of Lie groups) actually exist.
Later, Cartan solved two big problems. He classified all simple real Lie algebras and figured out how to represent Lie algebras in different ways. To do this, he came up with the idea of a "weight" for a representation.
In 1913, while studying how to represent orthogonal groups, Cartan discovered spinors. Spinors are mathematical objects that are very important in quantum mechanics, especially for describing particles like electrons.
After 1925, Cartan became more interested in the overall shape and structure (topology) of Lie groups. He showed that a connected Lie group can be thought of as a combination of a flat space and a compact group. He also found ways to understand the "holes" or "loops" (fundamental groups) in compact Lie groups by looking at their Lie algebras.
Exploring Differential Geometry
Cartan's work greatly improved differential geometry. This field studies curves, surfaces, and more complex shapes using calculus. Before Cartan, this area was becoming very complicated with many calculations.
Cartan used and greatly improved a method called "moving frames." This method helps to understand shapes by attaching a moving coordinate system to them. In modern terms, this idea is related to "fiber bundles," which are important in many areas of mathematics. Although Cartan didn't explicitly define fiber bundles, his work laid the groundwork for this concept.
He used his ideas to make Riemannian geometry (a type of geometry that deals with curved spaces) much clearer. His biggest contribution to Riemannian geometry was discovering and studying "symmetric Riemann spaces." These spaces have a special kind of "symmetry" around each point. Cartan showed that these spaces can be fully described using the classification of simple Lie groups. This connection means that symmetric Riemann spaces are important in other areas of mathematics, like analytic number theory.
Other Contributions
Cartan also developed a theory of gravity called Einstein–Cartan theory, which is an alternative to Einstein's general relativity.
See also
- Exterior derivative
- Integrability conditions for differential systems
- Isotropic line
- CAT(k) space
- Einstein – Cartan theory
- Hermitian symmetric space
- Moving frame
- Pseudogroup
- Pure spinor
Images for kids
-
A robot representing kid-friendly content.
