Cluster polylogarithms, Grassmannian polylogarithms and Zagier's conjecture on zeta_F(n), n >= 5

• Applicant: Dr. Steven Charlton
• Subject Area: Mathematics
• Term: from 2020 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 442093436

Zagier's Polylogarithm Conjecture is a promiment conjecture that relates special values of zeta functions (generalising the Riemann zeta function) with values of so-called poylogarithms (generalising the logarithm), and moreover connects number theory with algebraic K-theory.Goncharov has suggested a strategy for a possible proof of the Conjecture in higher weight. In particular, the so-called Grassmannian m-log (a multiple polylog in weight and depth m) is reduced to the classical polylog (in depth 1). So far the Conjecture can only be proven in weight <= 4 because of the combinatorial difficulties in and how little is known about the properties of multiple polylogs in higher weight. Recently Goncharov and Rudenko made a breakthrough that allowed them to prove the weight 4 case. They have discovered a deep new connection between polylogs and cluster varieties, which has greatly improved the understanding of weight 4 multiple polylogs.Any progress towards Zagier's conjecture for higher weight is inextricably linked with the better understanding of higher weight polylogs. This better understanding will benefit pure mathematicians working in number theory or algebraic K-theory, and theoretical physicists whose calculations of scattering amplitudes often invoke polylogs.The main goals of the project are:1. To define a family of cluster polylogs, whose cobrackets satisfy a recursive combinatorial formula. These functions should have good analytic properties and generalise Goncharov's and Rudenko's function L_4^1.2. To gain a good insight into the geometric functional equations of cluster polylogs. They should generalise Goncharov's and Rudenko's Q4 equation and give better candidates for our Q5 and Q6.3. To obtain a better expression for the 4-ratio through conceptual reduction of the Grassmanian 4-log with the help of geometric functional equations. Such a reduction is indespensible for higher weight, where explicit calculations are no longer practical.4. To reduce multiple polylogs of weight and depth n to those of depth n/2 with the help of cluster polylogs. This would confirm an important folklore conjecture.5. To find an expression for the Grassmannian m-log and its `coboundary' using cluster polylogs. This expression should then permit a reduction using geometric functional equations, in order to finally obtain the analogue of the cross-ratio in weight m.6. To construct a combinatorial model of the motivic Lie coalgebra using cluster functions and prove that it has the expected structure. It should be possible to do this explicitly in weight 5 by degenerating Q5. This should provide the crucial combinatorial step for a proof of Zagier's Conjecture in this weight.Supplementary: Continue to improve the efficiency of the computer implementation of the symbol calculation, to allow higher weight experimentation.

• DFG Programme: Research Grants