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 Δ 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 Δ 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 ∞ ∪ μ N , where μ N is the group of all N th 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.