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

zeta_F(n) 上的簇多对数、格拉斯曼多对数和 Zagier 猜想，n >= 5

• 批准号: 442093436
• 负责人: Dr. Steven Charlton
• 金额: --
• 依托单位: Max-Planck-Institut für Mathematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2020
• 资助国家: 德国
• 起止时间: 2019-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/442093436?language=en
• 关键词: Cluster; polylogarithms; Grassmannian; Zagier; conjecture

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.

扎吉尔的多对数猜想是一个 Proproment 猜想，它将 zeta 函数的特殊值（黎曼 zeta 函数的推广）与所谓的多对数（对数的推广）的值联系起来，此外还将数论与代数 K 理论联系起来。Goncharov 提出了一种可能在更高权重下证明该猜想的策略。 特别是，所谓的格拉斯曼 m-log（权重和深度 m 的多重多对数）被简化为经典多对数（深度为 1）。 到目前为止，该猜想只能在权重 <= 4 的情况下得到证明，因为组合困难，而且对较高权重的多个多对数的属性知之甚少。 最近，Goncharov 和 Rudenko 取得了突破，使他们能够证明重量为 4 的情况。 他们发现了多对数与簇变种之间的深刻新联系，这极大地提高了对权重4多重多对数的理解。Zagier关于更高权重猜想的任何进展都与更好地理解更高权重多对数有着千丝万缕的联系。 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，以及经常调用多对数计算散射幅度的理论物理学家。该项目的主要目标是：1。定义一个簇多对数族，其 cobrackets 满足递归组合公式。 这些函数应该具有良好的分析特性，并概括了 Goncharov 和 Rudenko 的函数 L_4^1.2。深入了解簇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 提供更好的候选方程。借助几何函数方程，通过概念性地简化格拉斯曼 4-log 来获得 4-比率的更好表达。 这种减少对于更高的重量是必不可少的，因为显式计算不再实用。4。借助聚类多对数，将权重和深度为 n 的多个多对数减少到深度为 n/2 的多对数。 这将证实一个重要的民间传说猜想。5．使用聚类多对数找到格拉斯曼 m-log 及其“共边界”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，以便最终获得权重 m.6 中交叉比的模拟。使用簇函数构建动机李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 中明确地做到这一点。 这应该为证明该权重的扎吉尔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，以允许更高的权重实验。