Dynamics, Statistics and Projective Geometry of Galois Fields

In the last paragraph of this book Arnold writes “The unity of mathematics is its main jewel. I have hoped here to contribute to this unity by the geometric presentation of Galois fields and of its relation to the ergodic theory of dynamical systems, to statistics, and to projective geometry, and also to returning all these forgotten classical theories to the continuous real world of the natural sciences.”

This wonderful little book is based on notes from a two hour presentation given to Moscow high school students. In just 80 pages it fulfills Arnold’s promise, making connections among a wide variety of mathematical fields. The table of contents and a preview are available at Google Books, but this doesn’t even begin to include the best parts.

Although originally aimed at the stupendously bright high school students mentioned above, I think this book would be better appreciated and understood by two other audiences. The first is professional mathematicians or graduate students with an interest in any of the book’s topics. In it they will find lovely connections among the topics, clear explanations, and much to think about.

The second audience would be students in an advanced undergraduate seminar-type course. The instructor would no doubt have to fill in missing pieces (the book is not self contained), and there are no exercises, but almost every sentence could be turned into an exercise. In addition the book illustrates mathematical research by numerical experiments leading to conjectures, most not yet proven. This approach will lead students to making their own discoveries, and if lucky, proofs, and the student is helped by several worked out examples from which to start their own investigations.

Arnold has done as much as is possible to achieve his stated goal, the rest is up to the readers of this book.

Peter Rabinovitch is a Systems Architect at Research in Motion, and a PhD student in probability. He fondly remembers learning differential equations from Arnold’s Ordinary Differential Equations.