Multiple polylogarithms, depth reductions and Zagier's conjecture

Polylogarithms (and the many variable generalisation, the multiple polylogarithms) are an important class of special functions which appear in many areas of pure mathematics (K-theory, number theory, hyperbolic geometry, differential geometry, ...) and high-energy physics (computation of Feynman integrals and of scattering amplitudes, ...).

I will give an introduction to the prominent results and conjectures on the structure of multiple polylogarithms, primarily originating with Goncharov, motivated by his programme to tackle Zagier's conjecture on special values \zeta_F(n) of the Dedekind zeta function.  I will then explain some of the/our recent results (involving collaborations of various subsets of myself, Andrei Matveiakin, Danylo Radchenko, Daniil Rudenko, and Herbert Gangl), wherein they/we establish identities which reduce the depth (number of arguments) of important combinations of multiple polylogarithms.  These results should be relevant for tackling Zagier's conjecture on \zeta_F(5) and \zeta_F(6).