MULTIPLE ZETA VALUES AND PERIODS OF MODULI SPACES (m)over-bar0,n

We prove a conjecture due to Goncharov and Manin which states that the periods of the moduli spaces m(0,n) of Riemann spheres with n marked points are multiple zeta values. We do this by introducing a differential algebra of multiple polylogarithms on m(0,n) and proving that it is closed under the operation of taking primitives. The main idea is to apply a version of Stokes' formula iteratively to reduce each period integral to multiple zeta values. We also give a geometric interpretation of the double shuffle relations, by showing that they are two extreme cases of general product formulae for periods which arise by considering natural maps between moduli spaces.