Henri Cartan has made monumental contributions in essentially every field of algebraic topology, including analytical functions, the theory of sheaves, homological theory, and potential theory. His most important works include Homological Algebra (1956) and Elementary Theory of Analytic Functions of One or Several Complex Variables (1963). He also worked on a definitive convergencetheorem on decreasing potentials of positive masses, which became a fundamental instrument for improving potential theory. Cartan is the recipient of the 1980 Wolf Prize in Mathematics.
Cartan was born in Nancy, France, in 1904. His father was Elie Joseph Cartan, a French mathematician who made significant contributions to the theory of subalgebras. Cartan was educated at the Ecole Normal Superieure in Paris, one of the finest schools for mathematics. Cartan attended the Ecole from 1923-1926. He was a protege of Jules Henri Poincare, who is considered the founder of algebraic topology and the theory of analytic functions of several complex variables. Along with Albert Einstein and Hendrik Lorentz, Poincare is considered a co-discoverer of the special theory of relativity. Cartan's studies with Poincare may explain his own interest in algebraic topology.
In 1935, Cartan was one of the founders of a group of young French mathematicians who had all graduated from the Ecole Normal Superieure. Other members included Claude Chevalley (or Chevallier), Jean Dieudonne, Jean Deslarte, and Andre Weil. All members were brilliant in their own right, however, Weil was the only "universalist" among them, accomplished in every area of mathematics. At the beginning, Cartan was not a universalist, although he did become more accomplished later in his career.
The group wrote under the pseudonym of Nicolas Bourbaki. In "Nicholas Bourbaki, Collective Mathematician," Claude Chevalley, as told to Denis Guedj, explains how Bourbaki got his name: "Weil had spent two years in India and for the thesis of one of his pupils he needed a result he couldn't find anywhere in the literature. He was convinced of its validity, but he was too lazy to write out the proof. His pupil, however, was content to put a note at the bottom of the page which referred to "Nicolas Bourbaki, of the Royal Academy of Poldavia'." The "real" Bourbaki liked to pretend he was in a secret society when he was young. From this, Weil tried to get the members of the group to stay anonymous. Members refused to answer questions about other members and of projects they were working, and even how the name of Bourbaki originated. The group did not remain anonymous for very long.
The Bourbaki group produced Elements de mathematique, a 30-volume textbook on analysis geared toward French university students, in 1939. It was intended to replace the class analysis textbook used in France written by Edouard Goursat. Bourbaki's book aimed to achieve the same high standards that Goursat set forth in his text, yet it also included mathematical advances. By 1968, there were 33 volumes in print. Early editions of Elements did not include credit to other contributors, so in 1960, Elements d'historique des mathematiques was published to rectify the situation.
The group sought the best way to produce the work together. They did not just want to assign topics to the person most qualified to write it, believing if they did, the book would turn out like an encyclopedia, which they did not want. Each member studied the same areas of mathematics, from the beginning. During the Bourbaki-congres, as their thrice yearly meetings were called, the group would discuss sections of the book, making numerous revisions. Bourbaki members had to retire from the group by the age of 50. This assurance was made to keep Bourbaki from "growing old," keeping him young in spirit. Women were not allowed to join the Bourbaki group. Through the Bourbaki members, a new kind of algebraic topology was born. Bourbaki played a key role in the rethinking of structural mathematics.
Cartan became president of the International Mathematical Union (IMU) in 1967, serving in that capacity until 1970. He then became the past-president, holding a seat on the executive committee for the following four years. The IMU is a scientific organization with the purpose of promoting international cooperation in mathematics and is part of the International Council of Scientific Unions. Founded in 1919, the IMU was disbanded in 1936, then reconstituted in 1951. Cartan was an invited speaker for a May 1995 IMU conference taking place at the Institut Henri Poincare.
Cartan's academic career began at the Lycee Caen in 1928, where he was a professor of mathematics. He left the Lycee in 1929, and was appointed a deputy professor at the University of Lille. From there, he went to the University of Strasbourg, which named him a professor of mathematics in 1931. From 1940 until 1969, Cartan was on the faculty at the University of Paris. After the University of Paris, Cartan taught at Orsay for about five years, until 1975.
In 1992, Cartan was part of a group of international scientists issuing a warning to humanity. The warning urged changes to protect the living world. According to the Worlds' Scientists Warning to Humanity, found on the World Wide Web, "Human beings and the natural world are on a collision course. . . . If not checked, many of our current practices put at serious risk the future that we wish for human society and the plant and animal kingdoms, and may so alter the living world that it will be unable to sustain life in the manner that we know. . . . Fundamental changes are urgent if we are to avoid the collision our present course will bring about." The warning lists several environments in need of help. Cartan's name is among the 1,500 signatories, all of which are members of regional, national, and international science academies.
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