## Armand Borel

### Mathematician whose work helps to explain why the range of physical materials is limited

©The work of the Swiss mathematician Armand Borel ranged from the purest mathematics, arithmetic and algebra, to theoretical work that underpins the understanding of sub-atomic particles. One of the world's leading mathematicians, he was a prolific worker who wrote nearly 200 papers -- collected in four volumes running to nearly 3,000 pages -- as well as 12 other books.Armand Borel was born in 1923 in La Chaux-de-Fonds, a small town in the French-speaking part of Switzerland. He graduated from the Federal Institute of Technology in Zurich in 1949, and took his doctorate at the University of Paris in 1952, the year he married Gabrielle Pittet.

In 1957 he moved as a professor to the Institute for Advanced Study at Princeton, where Einstein had worked and where Borel was to spend the rest of his career, until his retirement in 1993.

Borel joined a group of leading French mathematicians in the postwar era who were dissatisfied with mathematics and its foundations at that time, and sought to give the subject a new basis by rewriting it from scratch, emphasising clarity, rigour and the unity of the previously diverse branches of the subject. The group wrote collectively, under the pseudonym of Nicolas Bourbaki (who was a courageous but unsuccessful French general during the Franco-Prussian War). The many volumes of Bourbaki's Elements of Mathematics set new standards of exposition for mathematics, which have proved of lasting value.

Borel's own speciality was the theory of Lie groups. The concept of a group is an algebraic one, dating from the 19th century. The Norwegian mathematician Sophus Lie (1849-1929) was the first to make a systematic study of continuous groups of transformations (prototype: rotations, or motions of a rigid body). In this setting, there are three interrelated kinds of structure: from algebra (the group structure), from differential geometry (so one can apply the tools of calculus), and from topology (roughly, concerned with behaviour under continuous deformations).

Under suitable technical conditions, the interplay of these three types of structure restricts the possibilities so much that classification becomes feasible. Mathematicians are fascinated by such classification and structure theory. But so too are physicists. Lie groups are intimately related to the symmetries of nature, and their classification throws light on why there are only so many types of physical particle (of which atoms are built) -- and ultimately, only so many chemical elements (of which molecules, and all matter, are built).

In keeping with the unity of mathematics championed by Bourbaki, Borel contributed to other areas of pure mathematics as well. In particular he worked extensively on "arithmetic" problems -- group theory applied to algebraic number theory.

Borel was widely honoured, winning, for instance, the Steele Prize of the American Mathematical Society and being an invited speaker twice at the International Congress of Mathematicians (Stockholm 1962 and Vancouver 1974). He was probably most widely publicised, however, when he clashed with the Director of the Princeton Institute of Advanced Study in 1973. Borel and other scholars resisted plans to appoint a sociologist to a chair, questioning not only the credentials of the individual -- who did not, in the event, join the institute -- but also the validity of sociology as a discipline. In private life, Borel was a lover of music, like many mathematicians. He is survived by his wife and two daughters.

Armand Borel, mathematician, was born on May 21, 1923. He died of cancer on August 11, 2003, aged 80.

© The Times, 2003