This book derives from a 2-hour-long presentation to Moscow high-school students at the Moscow State (Lomonosov) University MGU, in November 2004. It is a translation from the Russian of The Dynamics, Statistics and Projective Geometry of Galois Fields, which was itself based on the earlier article Geometry and Dynamics of Galois Fields. It describes some astonishing recent discoveries of the relations between Galois fields, dynamical systems, ergodic theory, statistics and chaos, as well as of the geometry of projective structures on finite sets.

Most of these recent discoveries encapsulated empirical studies, and some of the conjectures suggested by these numerical experiments are still unproved, despite the fact that their simple statements make them quite accessible to high-school students (who can study them empirically, thanks to computers).

Together with these continuing empirical studies, it would be nice to investigate some of the remaining theoretical questions, such as the natural problem of the intrinsic characterisation of projective permutations among all the permutations of a finite set. We ought to be able to understand those geometrical features of some special permutations of a dozen points that make these special permutations projective, there by distinguishing them from non-projective permutations.

The author thanks the audience for many helpful remarks and hopes to extend the collaboration with the readers of the present book. The author looks forward to there being many contributions to this young domain of mathematics (including, one hopes, the discovery of applications of Galois fields beyond mathematics).