The center problem for the Abel equation, compositions of functions, and moment conditions

An Abel differential equation $y'=p(x)y^2+q(x)y^3$ is said to have a center at a pair of complex numbers $(a,b)$ if $y(a)=y(b)$ for every solution $y(x)$ with the initial value $y(a)$ small enough. This notion is closely related to the classical center-focus problem for plane vector fields. Recently, conditions for the Abel equation to have a center have been related to the composition factorization of $P=\int p$ and $Q=\int q$ on the one hand and to vanishing conditions for the moments $m_{i,j}=\int P^iQ^jq$ on the other hand. We give a detailed review of the recent results in each of these directions.