Would you like a pocket guide to the frontiers of modern research in pure mathematics? What fields are hot, which researchers stand out, what results will be viewed always as major achievements? The book under review is such a book.
The only previous book I would put in the same category is A Panorama of Pure Mathematics, as seen by N. Bourbaki , by Jean Dieudonné. Both books are very interesting, and I will describe Monastyrsky's new book by comparing it with Dieudonné's older and better-known book.
Neither book is for the faint of heart. Both are aimed at readers wanting a serious tour of pure mathematics. Both, as a practical matter, assume that the reader has the technical knowledge and mathematical maturity at least of a very good student finishing an undergraduate degree.
A driving force behind both books is the extremely ambitious desire to survey the entire landscape. To write a short book about the huge expanse of modern research, one has to make some hard choices as to what to include. Monastyrsky lets this choice be made by outside judges; he focuses on the achievements of the 38 Fields medalists through 1994. Dieudonné "yielded" to the judgement of a committee of which he was a co-founder; he focused on the mathematics covered by the 560 Bourbaki seminars through 1980.
A second driving force behind both books is the desire to present mathematics as a unified subject. Monastyrsky writes in his prologue "Mathematics is a single subject, a fact that is not always obvious when you study the daily reality of research." Dieudonné writes "One of the characteristics of Bourbaki mathematics is its extraordinary unity: there is hardly any idea in one theory that does not have notable repercussions in several others."
Monastyrsky's book consists only of some 160 rather small pages, and thus is about one-third the length of Dieudonné's 280 big-page book. One reason that Monastyrsky's book is short is that he tends to head straight for the spectacular results. In contrast, Dieudonné's book is encyclopedic in nature; he concentrates quite a lot on key definitions.
Monastyrsky's book obviously has a human focus, and this lends it a charm missing from Dieudonné's book. Monastyrsky often emphasizes work of the medalists done after the award too, and there are often striking changes of discipline. While there is hardly any specifically biographical information, the reader is given some feel for the mathematicians involved, not just the mathematics.
Monastyrsky is less judgemental than Dieudonné with respect to the relative importance of fields. Dieudonné is extremely clear. He rates fields from A to D, according to their "Bourbaki density". Some huge parts of mathematics don't even make the D class; combinatorics, for instance, is mentioned only as being a source of "problems without issue". Many readers will prefer Monastyrsky's consistently positive tone, especially when combined with his particular interest in connections between mathematics and physics.
Monastyrsky is also more of a diplomat with respect to the relative importance of individuals. He makes clear that he feels that the Fields committee has always chosen very worthy recipients. It is understood, however, that there is some arbitrariness in the process, and he appropriately singles out many other researchers for special mention. In particular, the point that pre-1990 Soviet mathematicians received fewer than their share of medals is made without assigning any particular blame. Dieudonné, on the other hand, is informative, but very heavy-handed. He gives a very short list of principal contributors to a given field, and then a longer list of secondary contributors.
I must say that Monastyrsky does not have quite the firm control over his material that Dieudonné had. For example, the Riemann hypothesis is just not relevant to Faltings' proof of the Mordell conjecture, while Monastyrsky suggests that its unproven status presented a crucial difficulty. In general, when reading Dieudonné, I often have the reaction "Wow, so much was faithfully captured in that sentence!" With Monastyrsky, I often have the reaction "Gee, close, but not quite hitting the nail on the head."
In terms of the craftsmanship visible in the final product, I again must say that Dieudonné's book is superior to Monastyrsky's book. The global organization of Monastyrsky's book is erratic, as sections written at different times have been simply laid side-by-side, rather than truly merged. For example, the 1990 Kyoto congress is in the future on page 5 and in the past on page 8. Even locally, Monastyrsky's book sometimes reads like a penultimate draft. For example, in the section centering on Selberg, an important distinction is obscured by using the notations c, c0, and c0' erratically.
This book marks Monastyrsky's second ambitious synthetic work, the first being Riemann, Topology, and Physics. Freeman Dyson wrote a positively glowing introduction to each of these two books. Riemann, Topology, and Physics was very well received, and is just now coming out in a corrected second edition.
I already recommend Modern Mathematics in the Light of the Fields Medals. I hope to fully share Dyson's enthusiasm when it too comes out in a corrected second edition.
David Roberts is currently an assistant professor of mathematics at Rutgers University. Starting in August 1999, he will be an assistant professor of mathematics at University of Minnesota, Morris.