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The Selected Works of Siguđur Helgason

G. Ólafsson and Henrik Schlichtkrull, editors
American Mathematical Society
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
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I recently reviewed a book entitled Geometric Analysis on Symmetric Spaces. I stated in that review that the book “is a model of fine scholarship and must rank as a definitive source for the indicated material.” Of course the author of the book was Sigurdur Helgason and I think it is fair say that this phrase qualifies, too, as a broad description of his work as represented in the volume I have the pleasure to review now.

Indeed, The Selected Works of Sigurdur Helgason, edited by Gestur Olafsson and Henrik Schlichtkrull, and published by the AMS, is a collection of thirty-two articles by Helgason (some with co-authors) spanning the years from 1956 to the present (2007, actually), and dealing with a wide range of topics including Banach algebras, Radon transforms, Paley-Wiener theorems in different settings, group representations, and Huygens’ principle. The papers are reproduced in their original journal form, which lends a certain ineffable charm to the older papers, given that nowadays journals’ appearances are nigh on homogeneous qua type-setting.

The Selected Works of Sigurdur Helgason, weighing in at over 700 pages, comes equipped with a preface by the editors, a biographical note, and an Introduction by Helgason himself, starting off with a charming brief description of his life, and followed by eight appraisals of the papers that follow as well as the author’s other works not included in this selection. These informative appraisals are grouped in accord with areas Helgason has worked in: invariant differential operators, geometric properties of solutions (of certain differential equations, e.g., the wave equation (in a symmetric space context), the Laplace equation, and so on), double fibrations in integral geometry and Radon transforms, spherical functions and spherical transforms, duality for symmetric spaces, representation theory, Fourier transforms on quotients G/K (again: symmetric spaces), and multipliers. (For personal mathematical reasons, I am personally most keen on the last four themes.)

These remarks by Helgason are in themselves representative of the author’s clear and incisive style, and, in addition to being a fine introduction to the papers in question, provide a view of how Helgason’s work fits into the broader context of twentieth (and twenty-first) century mathematical research in these areas.

After these remarks the book presents over a dozen nice photographs and a complete list, to date (actually, to 2008), of Helgason’s ninety-two published papers and eight books. Browsing the tiles of the books alone, and musing on the recognized impact they continue to have to this day, underscores the stature of Helgason as a scholar.

It is clearly fitting and proper that this volume should appear as a publication in this prestigious AMS series.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.