Index theory is one of those wonderful things in mathematics that ties together different areas and appears in many places and in many guises. It is of course fundamentally differential geometry, or algebraic topology, or a weighted linear combination of the two like differential topology, but it also appears in many other locales. The physicists are using index theory, of course, and this is certainly in keeping with the mathematical Weltanschauung of, for instance, Atiyah (one of the founders in the title of the book under review), but we find it also in analysis proper, in number theory, and … well, pick your own favorite example. Thus it is no exaggeration to describe index theory, centered on the famous Atiyah-Singer Index Theorem, as a mainstay of contemporary mathematical culture, and as both a hugely fecund technical theme and very elegant mathematics in its own right.
For experts as well as for fellow-travellers, the compendium of articles comprising The Founders of Index Theory is on target. The insiders are regaled by close-to-home reminiscences by (and about) the main players in the game, all of whom are household words to them. Interested outsiders (or tangential players, perhaps) are bound to be entertained and enlightened by what is in the pages of this book. In this day and age when history of mathematics is practiced at many different levels, the articles of the book will appeal also on this front. Finally, there is something particularly encouraging about accounts and analyses presented by old-fashioned (and stellar) scholars talking shop — or talking about talking shop — with each other: there is a refreshing quality of authenticity running throughout the articles, and there is a light touch present throughout, as typifies the argot spoken by friends to friends and about other friends. And there is a good deal of humor there, as well as poignancy.
The book is beautifully laid out, with a large number of nice photographs. It is brimming with articles by and about the “Founders,” i.e. Atiyah, Bott, Hirzebruch, and Singer, and it is all capped off by a section titled, “The ‘Gang of Four’ Together,” which is in itself a study in how dramatically successful cooperation flows from the interaction of these men.
Among the many fine, fine articles in the book we find such gems as Simon Donaldson’s discussion, “Geometry in Oxford, c. 1980–85”; Edward Witten’s study, “Michael Atiyah and the Physics/Geometry Interface”; Loren Tu’s “The Life and Works of Raoul Bott” and “Reminiscences of Working with Raoul Bott”; Nancy Hingston’s “Pure Thought” (how’s that for a provocative title?); some correspondence between J. A. Todd and Hirzebruch containing very pretty Mathematik; Richard Kadison’s “Which Singer is That? (another fetching title, no?); S-T. Yau’s “My Friendship with Singer”; and, at the end of the last chapter, a trio of irresistible articles: Ronald Douglas’ “The Evolution of Modern Analysis,” Lars Gårding’s “A Happy Collaboration,” and Allan Weinstein’s “Memories of the Gang of Four.” The book finishes with bibliographies of the wonderful scholars fêted in the preceding 350 pages or so.
For good measure and by way of whetting appetites, here is a bit from p.107, from Allyn Jackson’s interview with the late Raoul Bott.
Notices: Are you a geometric thinker? Do you visualize things a lot …?
Bott: My memory is definitely visual, but I also like formulas. I like the magical aspects of classical mathematics. My instincts are always to get as explicit as possible.
In most of my papers with Atiyah he wrote final drafts and his tendency was to make them more abstract. But when I worked with Chern, I wrote the final draft. Chern actually wrote a more down-to-earth version of our joint paper.
Notices: Is Chern even more of a “formulas man” than you are?
Bott: Oh, yes, I’m pretty bad, but he’s even worse! It’s strange that in some sense it was he who taught us how to work conceptually, but in his own work his first proofs are nearly always computational.
What a wonderful peek behind the scenes regarding these geometers’ methods!
And this pretty much describes this wonderful book from cover to cover.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.