Collection Dynamique et graphes sur les corps finis : aspects algebriques, arithmétiques et algorithmiques / Dynamics and graphs over finite fields: algebraic, number theoretic and algorithmic aspects
The field of dynamical systems generated by iteration of polynomials and rational functions is a classical area of mathematics with a rich history and a wide variety of results. Recently, there has been substantial interest in Arithmetical Dynamical Systems (ADS), meaning the iteration of rational functions over fields of number-theoretic interest. Although isolated results regarding ADS have been proven throughout the twentieth century, it was only in the 1990's that ADS were identied as a field of study. The past two decades have seen an explosion of work in this topic: fundamental problems have been solved, new questions and conjectures have been posed, connections have been forged with a great many different areas of pure and applied mathematics. Despite this recent profusion of scientic activity in ADS and related areas, little progress has been made on dynamical systems over finite algebraic structures such as finite fields (DFF) and rings (DFR). The proposed workshop will be a follow-up to the problems discussed at the workshop 'The Art of Iterating Rational Functions over Finite Fields', BIRS, Banff, May 2013, that was the first meeting to focus on DFF/DFR and their vast applications. There has been a recent burst in activity in the study of functional graphs generated by algebraic maps over finite fields. This comes from several independent groups of researchers who were not aware of each other's results prior to the aforementioned Banff workshop. It is expected that over the next two years we will witness strong growth of interest in this area, a score of new applications and diversication amongst the researchers who work on them. The goal of the suggested CIRM meeting will be to bring together the most active and distinguished researchers to allow them to exchange the ideas and set new coherent goals for future development of this exciting and versatile direction. The topics of the proposed workshop will center on the following closely related and cross-fertilising directions in DFF/DFR and functional graphs: Algebraic aspects Number theoretic aspects Graph theory aspects Algorithmic aspects We expect that the proposed workshop will open a floodgate for new ideas, problems, and methods as well as concrete results in this area. Furthermore, the problems proposed as the goals of this proposal have barely been touched upon by other meetings. On the other hand, we believe that there is now enough of a theoretical background and also active interest shown by many distinguished researchers in the area of DFF/DFR and related fields to successfully attack many of the long-standing problems. Bringing together researchers so that they can combine their techniques (and of course, intellectual efforts) will certainly lead to groundbreaking results.