Zagier's Polylogarithm Conjecture revisited

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 given. 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 impressionist talk I intend to give a rough idea of the developments from the early days on, avoiding most of the technical bits, and also hint at a number of recent results for higher weight (joint with S.Charlton and D.Radchenko).

About Herbert Gangl

My work is concentrated around polylogarithms and variants thereof, in connection with algebraic K-theory, algebraic number theory and arithmetic algebraic geometry. The questions that I am interested in lead to a variety of rather disparate topics, comprising for instance the homology of the general linear group, configuration and moduli spaces, combinatorial Hopf algebras, multiple zeta values or (quasi-)modular forms and in recent years particle physics. A good chunk of my research has a computer-experimental flavour, using mostly the GP/PARI scripting language to try to detect patterns in piles of data, to find hitherto unknown objects or to formulate conjectures.

https://www.durham.ac.uk/staff/herbert-gangl/