Abstract

Instigated by work of Borel and Bloch, Zagier formulated his Polylogarithm Conjecture in the late eighties and proved it for weight 2. After a flurry of activity and advances at the time, notably by Goncharov who provided not only a proof for weight 3 but set out a vast program with a plethora of conjectural statements for attacking it, progress seemed to be stalled for a number of years. More recently, a solution to one of Goncharov’s central conjectures in weight 4 has been found. Moreover, by adopting a new point of view, work by Goncharov and Rudenko gave a proof of the original conjecture in weight 4.   In this talk I intend to give a rough idea of the developments from the early days on, avoiding most of the technical bits, and, time permitting, also hint at a number of recent results for higher weight with new formulas for Grassmannian and Aomoto polylogarithms in terms of iterated integrals (joint with S.Charlton and D.Radchenko).