Formal paths, iterated integrals and the center problem for ordinary differential equations

We continue the study of the center problem for the ordinary differential equation started in [A. Brudnyi, An explicit expression for the first return map in the center problem, J. Differential Equations 206 (2004) 306–314; A. Brudnyi, On the center problem for ordinary differential equations, Amer. J. Math. 128 (2006) 419–451; A. Brudnyi, An algebraic model for the center problem, Bull. Sci. Math. 128 (2004) 839–857; A. Brudnyi, On center sets of ODEs determined by moments of their coefficients, Bull. Sci. Math. 130 (2006) 33–48; A. Brudnyi, Vanishing of higher-order moments on Lipschitz curves, Bull. Sci. Math. 132 (3) (2008) 165–181]. In this paper we present the highlights of the algebraic theory of centers.