# Armand Borel 1923–2003

Armand Borel, an internationally recognized mathematician whose work was fundamental to the development and formation of modern mathematics, died on August 11 in Princeton, New Jersey. Professor Borel, who was 80, was a Professor Emeritus at the Institute for Advanced Study, where he had been a member of the Faculty since 1957.

In 1991, Professor Borel received the American Mathematical Society’s Steele Prize for lifelong contributions to mathematics. The award citation noted that Professor Borel’s work "provided the empirical base for a great swath of modern mathematics, and his observations pointed out the structures and mechanisms that became central concerns of mathematical activity."

"In the course of amassing these astounding achievements," the award citation continued, "he placed the facilities of the Institute for Advanced Study at the service of mathematics and mathematicians, using them to foster talent, share his ideas, and facilitate access to recent developments through seminars and lectures. It is just simply not possible to cite a career more accomplished or fruitful or one more meaningful to the contemporary mathematical community."

Armand Borel’s mathematical work centered on the theory of Lie groups. Because of the increasingly important place of this theory in the whole of mathematics, Borel’s work came to influence some of the most important developments of contemporary mathematics. His first great achievement was to apply to Lie groups and homogenous spaces the powerful techniques of algebraic topology developed by Leray, Cartan, and Steenrod. In 1992, Professor Borel received the International Balzan Prize for Mathematics "for his fundamental contributions to the theory of Lie groups, algebraic groups and arithmetic groups, and for his indefatigable action in favor of high quality in mathematical research and of the propagation of new ideas."

After 1955, Professor Borel turned to algebraic groups, producing the classic paper* "Groupes lineaires algebriques,"* which represented a turning point in the history of the subject and which led to many important developments. At the same time, he was studying, and eventually solving (with H. Chandra and A. Baily), some of the most basic and difficult problems of the theory of arithmetic groups. Professor Borel also worked on the cohomology of arithmetic groups and its applications, as well as various aspects of new cohomological theories, automorphic forms, and the infinite-dimensional representation theory of real and *p*-adic Lie groups.

"A dominant feature of Borel’s scientific production," noted a 1993 article in the* Notices of the American Mathematical Society*, "is the systematic and conclusive character of his contributions to solving questions that are diverse, difficult, and always important. He wrote more than 145 articles before 1982, which were collected in a three-volume set published in 1983. A fourth volume of subsequent articles was published in 2001. But his presence in contemporary mathematics goes beyond his own mathematical production. Borel has played an eminent role as stimulator and propagator of new ideas in the international mathematical community. In particular, he has repeatedly initiated and participated in seminars and summer schools where important new techniques and results were brought forth." Armand Borel was born May 21, 1923 in La Chaux-de-Fonds, Switzerland. He received his *diplome* from the Swiss Federal Institute of Technology in Zurich in 1947 and his *doctorat d’etat* from the University of Paris in 1952. He served as an assistant (1947-49) and professor (1955-57; 1983-86) at the Swiss Federal Institute. He went to the Centre Nationale de la Recherche Scientifique in Paris in 1949, and then to the University of Geneva as a Professor of Algebra in 1950. From 1952-54, he was a member in the School of Mathematics at the Institute, and, in 1957, was appointed a professor at the Institute. He became professor emeritus in 1993.

A passionate music lover with broad-ranging musical interests, Professor Borel initiated a concert series at the Institute in 1985, and directed the program through 1992. The highly accomplished musicians he invited to perform at the Institute included the Indian flutist M.S. Shashank-Inidna, who gave a performance of Carnatic South Indian Music; the jaxx pianist Kenny Barron; the New York Camerata, who performed 20^{th}-century music; the Art Farmer Jazz Quartet; the Chamber Music at Lincoln Center Trio; the Howard Shaw Endellion String Quartet; and Grafelmusik.

Professor Borel served as a visiting lecturer at the University of Chicago (1954-55), and as a visiting professor at the Massachusetts Institute of Technology (1958, 1969), the Tata Institute of Fundamental Research in Bombay (1961, 1983, 1990), the University of Paris (1964), the University of California at Berkeley (1975), the University of Chicago (1976), Yale University (1978) and Tohoku University in Sendai, Japan (1990).

Most recently, Armand Borel was the main organizer of the multi-year Summer Program at the Center of Mathematical Science at Zhejiang University, Hong Kong. Professor Borel spent four months of three academic years, from 1999-2001, at Zhejiang University in order to set up the program. In addition, he was working in collaboration with Lizhen Ji of the University of Michigan at Ann Arbor, with whom he was completing a book due for publication in 2004.

In 1972, Professor Borel received an honorary doctorate from the University of Geneva, and in 1978 was awarded the Brouwer Medal of the Dutch Mathematical Society. A member of a number of scientific societies, he was elected to the American Academy of Arts and Sciences in 1976 and to the National Academy of Sciences in 1987. He was also a foreign member of the Finish Academy of Sciences and Letters, the French Academy of Sciences, the American Philosophical Society, and the Academia Europaea. He was an Honorary Fellow of the Tata Institute of Fundamental Research.

Professor Borel was an editor of *Annals of Mathematics* from 1962 to 1979. He was an editor of *Inventiones mathematicae* from 1979-93 and of *Commentarii Mathematici Helvetici*. He presented a Plenary Address at the International Congress of Mathematicians (ICM) in Stockholm (1962); served as a member of the Consultative Committee of the ICM in Moscow (1966); gave an Invited Address at the ICM in Vancouver (1974); and served as Chariman of the Consultative Committee of the ICM in Helsinki (1978). In addition, he gave invited lectures at numerous conferences all over the world.

Professor Borel loved nature and wildlife, and until his recent illness he was active athletically and hiked extensively. He was deeply concerned about the future of our environment.

Professor Borel is survived by his wife, Gabrielle Aline Borel, and daughters Dominique Odette Susan Borel and Anne Christine Borel, both of New York City.