In this article , a recording of a lecture delivered by the author on November 13, 2004 for schoolchildren of the Small Mekhmat of Moscow State University , amazing recently discovered connections of Galois algebraic field theory with the theory of dynamical systems chaos and statistics on the one hand and with the geometry of projective structures on sets of finite number of points on the other. Most of these new discoveries have been discovered experimentally, and the resulting hypotheses in many cases have not yet been proven, although their understanding and empirical verification are easily accessible to schoolchildren, especially those who own a computer. Waiting for inquisitive researchers and many theoretical questions - for example, Question about , what distinguishes the subgroup of projective permutations in the complete group of all permutations of a finite set , What are the special geometric properties of projective permutations of a dozen points? Distinct from non-projective[edit]