The work of I. M. Gel'fand on functional analysis, algebra and topology

This survey is timed to coincide with the sixtieth birthday of I. M. Gel'fand. The authors have confined themselves to those branches of mathematics in which he has been engaged during the last decade, and in the various branches the chronology of the articles covered by the survey is different. Gel'fand's research in the theory of group representations, which has lasted for thirty years, falls into several cycles; the majority of his results are widely known and were dealt with in the survey [1*] on the occasion of his fiftieth birthday. For this reason we deal here only with the results of the last ten years. His first articles on integral geometry appeared more than ten years ago, but this branch of mathematics is still in a formative phase. Because of this we include in the survey an outline of Gelfand's basic research in integral geometry, not exluding some that is comparatively old. Topology is a new branch of his scientific activity; all the work in this field was carried out between 1968 and 1973. In the preparation of the survey we had the help of I. N. Bernshtein, M. I. Graev, and D. A. Kazhdan.