On the Life and Work of S. Helgason

Abstract

This article is a contribution to a Festschrift for S. Helgason. After a biographical sketch, we survey some of his research on several topics in geometric and harmonic analysis during his long and influential career. While not an exhaustive presentation of all facets of his research, for those topics covered we include reference to the current status of these areas. Preface Sigurður Helgason is known worldwide for his first book Differential Geometry and Symmetric Spaces. With this book he provided an entrance to the opus of Elie Cartan and Harish-Chandra to generations of mathematicians. On this the occa- sion of his 85th birthday we choose to reflect on the impact of Sigurður Helgason's sixty years of mathematical research. He was among the first to investigate system- atically the analysis of differential operators on reductive homogeneous spaces. His research on Radon-like transforms for homogeneous spaces presaged the resurgence of activity on this topic and continues to this day. Likewise he gave a geomet- rically motivated approach to harmonic analysis of symmetric spaces. Of course there is much more - eigenfunctions of invariant differential operators, propagation properties of differential operators, differential geometry of homogeneous spaces, historical profiles of mathematicians. Here we shall present a survey of some of these contributions, but first a brief look at the man.