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

Zagier的多对数猜想是将zeta函数的特殊值（推广黎曼zeta函数）与所谓的多对数值（推广对数）联系起来的一个预测猜想，并且将数论与代数k理论联系起来。Goncharov提出了一个在更高权重下证明猜想的策略。特别是，所谓的格拉斯曼m-log（权重和深度为m的多重多元对数）被简化为经典多元对数（深度为1）。到目前为止，这个猜想只能在权值<= 4的情况下被证明，因为在更高权值下的组合困难以及对多重多元曲线的性质所知甚少。最近，贡恰罗夫和鲁登科取得了一项突破，使他们能够证明体重4的说法。他们发现了多聚材料和簇状品种之间的深层新联系，这大大提高了对重4多聚材料的认识。对于更高权值的Zagier猜想的任何进展都与更好地理解更高权值的多元曲线密不可分。这种更好的理解将有利于研究数论或代数k理论的纯数学工作者，以及计算散射振幅经常使用多元曲线的理论物理学家。该项目的主要目标是：1。定义一类聚类多对数，其余括号满足一个递归组合公式。这些函数应具有良好的解析性质，并推广Goncharov和Rudenko的函数L_4^1.2。为了更好地理解聚类多元曲线的几何泛函方程。他们应该推广Goncharov和Rudenko的Q4方程，为我们的Q5和Q6.3提供更好的候选方程。利用几何泛函方程对格拉斯曼4对数进行概念化简，得到4比的更好的表达式。这样的减少对于更高的重量是必不可少的，因为明确的计算不再实用。利用聚类多元曲线将权重和深度为n的多元曲线简化为深度为n/2的多元曲线。这将证实一个重要的民间传说猜想。用聚类多对数找到格拉斯曼m-对数及其“共边界”的表达式。然后，该表达式应允许使用几何泛函方程进行约化，以便最终获得重量m.6的交叉比的类似物。利用聚类函数构造动机李共代数的组合模型，并证明其具有预期的结构。通过退化Q5，应该可以在权重5中显式地做到这一点。这将为在这个权重中证明Zagier猜想提供关键的组合步骤。补充：继续提高计算机执行符号计算的效率，以允许更高权重的实验。

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.