Poincaré was, according to E. T. Bell, “the last universalist,” a scholar who made all of mathematics, pure as well as applied, his province. And his contributions to an unimaginably wide variety of fields were often of a seminal level. One has only to think of his youthful competition with Klein in connection with the overture to what is now the theory of automorphic functions, his founding of the subject of algebraic topology (are there any other contenders?), or even his work in theoretical physics. In this connection, recall for example the famous photograph taken at the First Solvay Congress, where we find Poincaré and Madame Curie huddled over a paper, flanked by, among others, Lorentz, Rutherford, Kamerlingh-Onnes, and a very young Einstein.
The book under review characterizes Poincaré as a “universal specialist,” citing. e.g, his role (in the guise of a mining engineer) in the investigation of a firedamp explosion in the Magny mineshafts, his assistance to Heinrich Hertz in computing the speed of a wave passing through a twisted wire, his supervision of new measurements of Quito’s meridian arc, and even his part as an expert witness in the notorious Dreyfus case (cf. Émile Zola’s famous J’Accuse!). Perhaps only John von Neumann can approximate this kind of a resumé, but it is clear that qua breadth of mathematical activity Poincaré is in a class with Euler, Gauss, Riemann, and Hilbert (who might be offered as a Gegenbeispiel to Bell’s claim, mentioned above).
The Scientific Legacy of Poincaré is, accordingly, a compendium of expository articles, by a number of different authors, on the labors of this wonderful mathematician. The book, edited by Éric Charpentier, Étienne Ghys, and Annick Lesne, covers in nineteen chapters nineteen different aspects of scholarship, all of major importance (modulo certain variances, of course). So it is that Ch’s. 2, 5, and 13 deal with Poincaré’s work on differential equations, with the latter chapter hitting PDE; Ch. 3 is a superb discussion of Poincaré series; Ch 11 addresses one of the most important themes in mathematics, namely the concept of residue (and, yes, this is where (co)homology takes the stage) and then, in short order, we get in Ch. 12 a discussion of the Poincaré conjecture featuring none other than Sasha Perelman. Thereafter, following discussions of Poincaré and probability, we get (Ch. 16) Poincaré and Lie theory, (Ch. 17) the Poincaré group (in physics), and, in the last two chapters, appraisals of Poincaré as an applied mathematician and a philosopher of science.
The articles are very well written, indeed, and are of course autonomous. But even non-specialists will want to sample these wares. The mathematics is presented clearly and very accessible, and the numerous historical accounts and asides make add an additional welcome cultural element to whole experience.
The Scientific Legacy of Poinaré is bound to be a hit across the mathematical spectrum: it has something for every one interested in any aspect of Poincaré’s work, which is to say, something for every one.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.