Goncharov's programme, and depth reductions of multiple polylogarithms (in weight 6)

Multiple polylogarithms $Li_{k_1,\ldots,k_d}(x_1,\ldots,x_d)$ are a class of multi-variable special functions appearing in connection with K-theory, hyperbolic geometry, values of zeta functions/L-functions/Mahler measures, mixed Tate motives, and in high-energy physics.

One of the main challenges in the study of multiple polylogarithms revolves around understanding how on many variables a multiple polylogarithm function (or 'interesting' combinations thereof) actually depend (''the depth''), as for example $Li_{1,1}$ can already be expressed via $Li_2$. Goncharov gave a conjectural criterion (''the Depth Conjecture'') for determining this, using the motivic coproduct, as part of his programme to investigate Zagier's Polylogarithm Conjecture on values of the Dedekind zeta function $\zeta_F(m)$.