Galois symmetries of fundamental groupoids and noncommutative geometry

We define a Hopf algebra of motivic iterated integrals on the line and prove an explicit formula for the coproduct Delta in this Hopf algebra. We show that this formula encodes the group law of the automorphism group of a certain noncommutative variety. We relate the coproduct A to the coproduct in the Hopf algebra of decorated rooted plane trivalent trees, which is a plane decorated version of the one defined by Connes and Kreimer [CK]. As an application, we derive explicit formulas for the coproduct in the motivic multiple polylogarithm Hopf algebra. These formulas play a key role in the mysterious correspondence between the structure of the motivic fundamental group of P-1-({0, infinity} boolean OR mu(N)), where mu(N) is the group of all Nth roots of unity, and modular varieties for GL(m) (see [G6], [G7]). In Section 7 we discuss some general principles relating Feynman integrals and mixed motives. They are suggested by Section 4 and the Feynman integral approach for multiple polylogarithms on curves given in [G7]. The appendix contains background material.