Customer Reviews

Modern Mathematics in the Light of the Fields Medals

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Most professional mathematicians know next to nothing about branches of mathematics outside their own narrow specialty. If this describes you, then you *must* read this book. This beautiful little book of Monastyrsky gives a brilliant exposition of the work of all the Fields medalists up to and including the 1994 winners. It seems impossible in such a small amount of space to assume no more than what the average mathematician can be expected to know, and yet at the same time to provide enough technical detail for the reader to gain an accurate understanding of the content and significance of the major theorems of all the medalists. Perhaps this is actually impossible, but Monastyrsky comes as close as is humanly possible to achieving this goal. This book will broaden your mathematical culture more than any other single book I know. It was not until I read this book that I learned how simply the exotic structures on S^7 may be described, or what Margulis got his Fields medal for.

Warning: if you do not have at least a graduate-level education in mathematics, most of the book will be incomprehensible. Although there are some historical notes and insights into people's personalities scattered throughout the book, the intended audience is unquestionably the research mathematician who wants to know more than just the buzzwords associated with each Fields medalist.

So what is mathematics anyway? You may think you know, especially if you are a math grad student, but unless you're familiar with the contents of this book, you don't!

This book was a real eye-opener for me. Basically it covers what is considered to be important mathematics by the math community at large by recounting the discoveries/creations of the Fields Medalists. I found it fascinating how the Fields Medalists' work tied together large areas of mathematics together, and how many times this intertwining nature of their work wasn't realized until years later!

This book is rather incomprehensible initially, even if delightfully put together. The first time I read it, I couldn't pronounce some of the words. The second time, I knew what some of those words meant. The third time, I saw how the words I was comfortable with were related with those I weren't. One can read sections over and over again, each time having gained more knowledge of the mathematics involved, and still get a handy pointer on what to learn next. I think that is the greatest thing about the book: one can see the relation of what one is doing to the Grand Plan of mathematics and how the latter developed and is growing even now.

I've only gone over the topology section in some depth, since that is my area, but I've found it useful for pointing me towards what to concentrate on. The bibliography is very useful in that regard; I feel it could be more extensive, but certainly it does an admirable job in listing some of the more useful references.

Dyson's introduction describes this book as a "roadmap". Seen that way, you should get plenty of use out of it over a long period of time, although perhaps not immediately.

As a first-year graduate student, I undersood most topics tangentially at best. Nonetheless, I was able to get a feel for some of the last century's most contributions to all of mathematics. I whole-heartedly recommend this book to anyone with a lot of exposue to various fields of mathematics, even if your knowledge of those fields is limited to the "exposure" level.

This book, within 150 pages, not only attempts to describe the efforts of the mathematical Fields Medal winners up until 1994, but actually tries to use it as a compass to assess where the field of mathematics is going. Not an easy task, even for this mathematical polymath. It is rare for professional mathematicians to try to put into plain English what they are working on, but the author makes just such an effort, and I think it is a needed one.

The book attempts to summarize, in little over a hundred pages, the accomplishments of the Fields Medalists from the institution of the prize to 1994. Anyone familiar with the mathematics of the last century will understand the difficulty the author faced, but will recognize that even partial success is an accomplishment. Partial success is exactly what happened.

After reading this book, one will have a much better understanding of how a good deal of modern mathematics fits together. They will understand how Thurston's conjecture on geometrization of 3-manifolds relates to Serre's work on spectral sequences and Smale's work on the Poincare conjecture, for instance. The book is also laudable for a description of the Fields Medal award process, which (if it were read by the right folks) ought to dispel some nasty rumors about discrimination between sub-disciplines.

One caveat and one weakness should be observed. The caveat is that the text assumes enormous knowledge of pretty much all branches of mathematics. It is profitable reading even if you don't understand much, but be prepared. The weakness is one of perspective. The author gives considerable attention to anything bearing on physics, and not so much to anything else. That aside, the coverage is as full, accessible, and balanced as anyone can expect, and probably more so. Then again, that's a low standard.