p-adic multiple zeta values and p-adic pro-unipotent harmonic actions

Multiple zeta values are periods of the pro-unipotent fundamental groupoid of the projective line minus three points. We will explain a way to compute their p-adic analogues, which keeps track of the motivic Galois action, and which has an application to the finite multiple zeta values recently studied by Kaneko and Zagier. The computation will be expressed by means of new objects which we will call p-adic pro-unipotent harmonic actions.