Computation of p-adic multiple zeta values and motivic Galois theory

Multiple zeta values can be written as sums of series and as integrals. Their integral expression makes them into periods of the pro-unipotent fundamental groupoid of $\mathbb{P}^{1} - {0,1,\infty}$. p-Adic multiple zeta values are defined as p-adic analogues of these integrals. We will show how to express them as sums of series, which allows in particular to compute them explicitly. We will mention the role of finite multiple zeta values defined by Kaneko and Zagier, and of a question asked by Deligne and Goncharov on a relation between the computation of p-adic multiple zeta values and their algebraic properties. To express the results we will introduce new objects in relation with motivic Galois theory of periods.