Armand Borel

Switzerland/USA

1992 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 favour of high quality in mathematical research and the propagation of new ideas.

The considerable mathematical production of **Armand Borel** (*1923 - †2003) is centered on the theory of Lie groups (in its widest sense). By its numerous aspects and applications, that theory has taken an increasingly important place in the whole of mathematics; as a result, Borel’s work has influenced some of the most important developments of contemporary mathematics in a major way.

Borel’s first great achievement was the application to Lie groups and to homogeneous spaces of the powerful techniques of algebraic topology developed during and just after World War II by J. Leray, H. Cartan, N. Steenrod and others; of the very rich, quasi-exhaustive information we have at our disposal on the cohomological and homotopic invariants of Lie groups and symmetric spaces, a large part is due to Borel (and collaborators).

After 1955, he turned his attention to algebraic groups. His classical paper “Groupes linéaires algébriques” (Annals of Math., 1956) is a turning point in the history of the subject. In particular, it made possible the classification (by C. Chevalley) of semi-simple groups over any algebraically closed field. Later on, the “relative” theory, of great importance for the applications, the rationality problems and the study of “abstract” isomorphisms have been the topic of several papers (together with J. Tits and T.A. Springer).

At the same time, Armand Borel was studying and eventually solving (with Harish-Chandra and W. Baily) some of the most basic - as well as difficult - problems of the theory of arithmetic groups: reduction theory, cocompactness criteria, compactification of quotient spaces, etc.

During the last twenty years, Borel has also worked on cohomology of arithmetic groups, its applications (to K-theory etc.), various aspects of new cohomological theories (L2-cohomoiogy, intersection cohomology), as well as on the theory of automorphic forms and the infinite dimensional representation theory of real and p-adic Lie groups. To each one of these domains, he contributed in a significant way.

A dominant feature of Borel’s scientific production is the systematic and, so to speak, final character of his multiple contributions to questions which are extremely diverse, difficult and always important.

Borel wrote more than 145 articles and those which appeared before 1982 have been collected in 3 volumes of the “Collected Papers” and published, in 1983, by Springer-Verlag. Borel is also the author of the following works:

- Ensembles fondamentaux pour les groupes arithmétiques et formes automorphes, annotazioni di un corso tenuto all’Inst. H. Poincaré 1964, redatte da H. Jaquet, J.-J. Sansuc e B. Schiffmann, Ecole Normale Supérieure, Parigi 1967;

- Introduction aux groupes arithmétiques, Actualités Sci. md. no. 1341, Hermann, Parigi 1969;

- Linear Algebraic groups (Annotazioni di H. Bass), Math. Lecture Notes Series, Benjamin Inc. New York, 1969; Y edizione interamente riveduta e ampliata, Springer-Verlag 1991;

- Représentations de groupes localement compacts, Lect. Notes Math. 276, 1972.

But Armand Borel’s presence in contemporary mathematics goes beyond his own mathematical production, important as that is. Helped by his privileged position at the Institute for Advanced Study, and by an exceptional working capacity, he has played an eminent role of stimulator and propagator of new ideas in the international mathematical community. In particular, he has repeatedly been the initiator and key participant in seminars or summer schools where important new techniques and results were exposed at the highest level. Thanks to his indefatigable activity, several of those seminars produced volumes of proceedings - to which he personally contributed - which have become, by now, basic reference works for the subjects in question. Among them, one can mention:

- Seminar on transformation groups (Princeton, 1958-59), Annals of Mathematical Studies no. 46, Princeton University Press, 1960;

- Algebraic groups and discontinuous subgroups (A.M.S. Summer School, Boulder, 1965, ed. A. Borel - G.D. Mostow), Proc. Symposia Pure Math., no. 9, 1966;

- Seminar on algebraic groups and related finite groups (Princeton, 1968-69), Springer Lecture Notes in Math. no. 131, 1970;

- Automorphic forms, representations and L-functions (A.M.S. Summer School, Corvallis, 1977, ed. A. Borel - W. Casselman), Proc. Symp. Pure Math., no. 33, 1979;

- Continuous cohomology, discrete sub groups and representations of reductive groups (Princeton, 1978-79, ed. A. Borel - N. Wallach), Annals of Math. Studies no. 94, Princeton University Press, 1980;

- Intersection cohomology (Berna, 1983), Progress in Math. vol. 50, Birkhäuser Verlag, 1984;

- Algebraic D-modules (A. Borel ed altri), Perspectives in Math. no. 2, Academic Press, 1987.

(1992)