Abstract: A system of ordinary differential equations on the plane is said to have a center at one of its singular points if all its trajectories around this point are closed. It is a classical problem to give explicit necessary and sufficient conditions for a system to have a center (Center-Focus problem). Another closely related question is to count isolated closed trajectories of a plane system (second part of Hilbert's 16-th problem).

Recently in both these problems new connections have been found with some questions in classical analysis and algebra. In particular, this concerns the vanishing problem of certain moment-like expressions and of iterated integrals on one side, and the structure of the composition factorization of analytic functions on the other.

We present an overview of some recent developments in this direction.