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

Zagier的多重对数猜想是一个promiment猜想，它将zeta函数的特殊值（推广了Riemann zeta函数）与所谓的多重对数的值（推广了对数）联系起来，并且将数论与代数K理论联系起来。Goncharov提出了一种策略，可以在更高的权重下证明该猜想。 特别地，所谓的格拉斯曼m-log（在重量和深度m上的多个polylog）被简化为经典的polylog（在深度1上）。 到目前为止，由于组合上的困难以及对更高权的多重多对数的性质知之甚少，该猜想只能在权<= 4的情况下被证明。 最近Goncharov和Rudenko取得了突破，使他们能够证明重量4的情况。 他们发现了polylog与簇之间的深层新联系，这极大地提高了对权重为4的多重polylog的理解。任何对更高权重的Zagier猜想的进展都与更好地理解更高权重的polylog密不可分。 这种更好的理解将有利于从事数论或代数K理论工作的纯数学家，以及理论物理学家，他们对散射振幅的计算经常调用polylog。该项目的主要目标是：1.定义一族簇多元对数，其余括号满足递归组合公式。 这些函数应具有良好的解析性质，并推广了Goncharov函数和Rudenko函数L_4 ^[1.2]。深入研究丛集多对数的几何函数方程。 他们应该推广Goncharov和Rudenko的Q4方程，并为我们的Q5和Q6.3提供更好的候选者。利用几何函数方程对Grassmanian 4-log进行概念化约，得到更好的4-比表达式。 这样的减少对于更高的权重是不可缺少的，其中显式计算不再实用。利用多对数聚类方法将权值和深度为n的多对数约简为深度为n/2的多对数。 这将证实一个重要的民间传说。使用群集多对数找到格拉斯曼m-log及其“上边界”的表达式。 然后，该表达式应允许使用几何函数方程进行简化，以便最终获得重量m.6的交比的模拟。利用簇函数构造运动李余代数的组合模型，并证明其具有预期的结构。 应该可以通过退化Q5在权重5中明确地做到这一点。 这应该为证明Zagier猜想提供了关键的组合步骤。补充：继续提高符号计算的计算机实现效率，以允许更高的权重实验。

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.