#### Distinguished French mathematician and member of Bourbaki

Henri Cartan, who has died aged 104, was one of the world's most distinguished mathematicians, his life's work conducted at the frontiers of pure mathematics, especially analytic functions and algebraic topology. His students included some of the greatest mathematical minds of the 20th century. He was also a member of Bourbaki - a collective pseudonym for a group of young, mainly French, mathematicians whose books caused a quiet revolution in mathematics from the 1930s onwards.

Born in Nancy, north-eastern France, the son of the mathematical giant Élie Cartan and his wife Marie-Louise Bianconi, Henri had two brothers and a sister. Family life was intellectual, with a strong emphasis on music. Élie tried not to influence his children's choice of career, but Henri developed an early interest in mathematics. He went to the prestigious École Normale Supérieure in Paris, where he met another outstanding young mathematician, André Weil, and attended mathematics courses given by his father and by Gaston Julia.

His doctoral supervisor was Paul Montel, who did research on analytic functions. This core area of mathematics had arisen in the 19th century as a fusion between analysis - a logically rigorous development of the calculus of Isaac Newton and Gottfried Leibniz - and complex numbers, in which negative numbers possess square roots. Although such ideas may seem rather strange, they quickly turned out to be vital, not just to mathematics but to many of its applications, including fluid mechanics and electromagnetism. Montel's interest, like the classical work that led up to it, was in functions of a single complex variable. That is, rules, usually embodied in formulas such as "take the square root", for deriving one complex number from another.

Weil suggested to Cartan that it might be interesting to extend these ideas to several complex variables, an innocent-sounding generalisation that turned out to have unexpected depths. Most of the obvious guesses about how the classical theory should extend to many variables turned out to be wrong. Cartan followed his friend's advice, and, in 1931, he published a joint paper with his father, combining his own interests with Élie's expertise in Lie groups - a far-reaching theory of symmetry.

In 1935 Cartan joined Bourbaki, the mathematical collective that originated as an attempt to fill the vacuum created by the deaths of many leading mathematicians in the first world war. The pseudonymous Bourbaki published a string of classic textbooks. Their strengths were logical organisation and careful formulation of key concepts; their main weakness was a tendency to develop ideas at a forbidding level of abstraction and generality.

Also in 1935, Cartan married Nicole Weiss, and together they had two sons and three daughters. In 1939 he was teaching at the University of Strasbourg, which was evacuated when the second world war began. By the following year, he was at the Sorbonne, in Paris, where he spent much of his career. Cartan's brother Louis joined the resistance, and was executed by the Nazis in 1943. Cartan visited the US to work with Samuel Eilenberg, and their 1956 book

*Homological Algebra*is still well known (as Cartan and Eilenberg) and widely used. This marked a new strand in Cartan's research, moving into the rapidly growing area of topology.

Topology is geometric in viewpoint, but it deals with questions that Euclid never thought of. In topology, triangles and circles are identical, because each can be continuously deformed into the other. Topology is about features that survive such deformations: links, knots, holes. The subject took off when these intuitive notions were given specific mathematical meanings. It quickly turned out to be of fundamental importance, because the key concept of continuity is crucial to many parts of mathematics. Today's quantum physics has become a major consumer of topological ideas. One motivation for Cartan's interest in topology was its relevance to analytic function theory. "Several variables" is an analytic way to express the geometry of spaces with many dimensions, a point of view that was beginning to pervade the whole of mathematics. Many of the world's most prominent mathematicians attended Cartan's seminars at the Sorbonne - among them Jean-Pierre Serre and Alexander Grothendieck, recipients of Fields medals, the mathematical community's highest award. Cartan had few official graduate students, but these included Serre, René Thom and Adrien Douady. Thom became famous in the 1970s for catastrophe theory, a deep mathematical analysis of singularities whose potential applications controversially included biology and social science. Douady made major contributions to fractals - complex shapes that have intricate structure however much they are magnified, which have many applications in science and finance.

Cartan was also a human rights campaigner, masterminding the release of the mathematician Leonard Plyushch from a Soviet psychiatric hospital in 1976, and campaigning on behalf of other mathematical dissidents such as the Uruguayan José Luis Massera. He received numerous honours, including the gold medal of CNRS (the French National Centre of Scientific Research) in 1976 and the Wolf Prize in 1980. He was appointed Commandeur de la Légion d'honneur in 1989.

**Ian Stewart**

*Henri Paul Cartan, mathematician,*8

*born July*1904;

*died*13

*August*2008

1 October 2008 © Guardian Newspapers Limited 2008