Multiple zeta values and multiple Hurwitz-zeta values. Analytic and combinatorial aspects

In this talk, we shall discuss about some new results in the evaluation of some multiple zeta values (MZV). After a careful introduction of the multiple zeta values (Euler-Zagier sums) we point out some conjectures back in the early days of MZV and their combinatorial aspects. Moreover, we provide an evaluation of the Hoffman basis in terms of an infinite rational series involving even values of the Riemann zeta function. We also discuss similar results for the multiple t-values and alternating multiple zeta values. The multiple zeta values (Euler-Zagier sums) were introduced independently by Hoffman and Zagier in 1992 and they play a crucial role at the interface between analysis, number theory, combinatorics, algebra and physics.