Ritt decompositions --- the known, unknown, and their applications
Many properties of rational functions f arising from problems in number theory, dynamics, and complex analysis, can be studied by writing f as a composition f_1 o ... o f_r, where the f_i's are indecomposable rational functions, i.e. cannot be decomposed nontrivially further. However basic questions such as determining the relationship between two such decompositions of f remain unknown. We shall describe progress towards a description and its applications to various problems.