There are at least two groups of mathematicians for whom the book under review is bound to be all but irresistible: insiders with an interest in how their subject has evolved over the last quarter century or so — in view of the state of the art in 1975 —, and outsiders who have a need for a road map of sorts into at least major parts of this very highly developed subject. Since 1954, the issuing agency, Centro Internazionale Matematico Estivo, has held summer courses on certain mathematical themes taught by marvelous experts: the present offering on differential operators on manifolds features Atiyah, Bott, Stein, Malliavin, and Helgason. The lengths of these de facto mini-course notes average around 50 pages or so, and accordingly the reader can expect a non-trivial amount of consciousness-raising throughout: what a wonderful way to learn about these topics. And, as I already indicated, the more expert reader can revel in the mastery of the authors as they present these beautiful subjects in a purposely expository fashion, fitted into the perspective of over 30 years ago.
The presumed level of sophistication of the reader can be gauged by e.g., the introductory remarks by Atiyah and Bott: “The aim of [Atiyah’s] first two lectures is to review some basic facts about the representation theory of the unitary group … and the special orthogonal group …” and [so sagt der Bott:] “A central theme in the development of topology and geometry has been the relation between Riemannian curvature and the topological shape of a manifold … [To wit:] certain universal polynomials in the curvature tensor R of a Riemannian structure g give rise to differential forms [… the Pontryagin forms of g] … whose cohomology classes in the de Rham theory of M [the manifold] are independent of g … My first aim in these lectures will be to present, in a self-contained manner, a relatively new way of getting at these classes …” After these introductions the authors get air-borne rather quickly and go on to cover a lot of beautiful scenery.
Therefore, given that they tend to soar rather high, preliminary work in differential geometry, e.g. a course on differentiable manifolds with a heavy emphasis on de Rham theory (unavoidable, really) and a course on Riemannian geometry (or both in one) is really a non-negotiable prerequisite, unless the reader is keen on learning a lot of this material (nonlinearly) in the context of the more arcane and advanced material the authors address in their articles. And I guess this latter proviso opens the door to the possibility of using the articles in the book under review as supplements to other studies, perhaps carried out simultaneously. For example, if one has a keen interest in characteristic classes, this kind of “both ends against the middle” approach could be quite fecund. (For what it’s worth this is the boat I am in myself.)
The articles by Stein and Helgason can probably be characterized as more analytic and less geometrical than the preceding ones by Atiyah and Bott, which is no surprise, of course. It follows that they are of a very different flavor than their predecessors. It thus comes to pass that Stein trains his focus on singular integrals and Helgason addresses symmetric spaces in a Lie theoretic setting. Malliavin’s article is also apparently more analytic (and potential theoretic, and global analytic …) than those by Atiah and by Bott.
I am unquestionably a member of the group of interested outsiders mentioned in the first paragraph above: I very much look forward to reading and even studying some of these exposés with some care, prepared to fill the margins of Differential Operators on Manifolds with notes, questions, and what not. If there is no royal road to geometry, as King Ptolemy was allegedly told by Euclid (though I recall first hearing the legend about Alexander the Great, whose tutor (and interlocutor in this tale) was Aristotle), the present path should still be less taxing than others: with such masters as these in the game, and with the exposition being this elegant, lucid, and cogent, it should all be pure pleasure.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.