Monodromy action and limit cycles 

The study of the monodromy action associated to a differential equation given by an al- gebraic vector field is an approximation to the 16-Hilbert problem. The latter consists of, considering a polynomial vector field in the plane, determining a bound for the number of limit cycles. This problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900, and remains an open problem. However, there are several cases in which we may have some partial solutions to the problem. In this talk, we will discuss how by thinking about a weaker version of the 16-Hilbert problem, the characterization problem of when an Abelian integral is exactly zero naturally arises. This problem is known as the tangential-center problem, and its most notable results are based on the study of the monodromy action associated with certain polynomials.