Multiple Zeta Values in Function Fields using Motivic Framework

多个zeta值是实数，由某些分数的无限和定义。自欧拉时代以来，它们的确切值和它们之间的关系一直困扰着数学家。特别是，数学家试图理解何时可能将有限数量的多个zeta值加在一起以得到另一个多个zeta值。虽然数学家有无数这样的关系的例子，但证明这些构成了所有这样的关系仍然是一个遥远的目标。这个项目将研究多个zeta值在另一种设置中的近亲，在这种设置中有一个合理的希望来证明关于它们之间所有关系的集合的这样的结果。该项目的最终目标是开发一种新的方法来产生多个zeta值之间的关系族，并对这种方法产生的关系进行分类。该项目还将与附近的hbcu进行接触，以鼓励代表性不足的群体参与研究生水平的数学和STEM项目。更具体地说，这个项目的主要目标是为证明在全局函数域上定义的多个zeta值的某些集合的代数独立性奠定基础。第一步是开发一种生成函数域上定义的多个zeta值之间关系的方法。PI将使用他最近开发的公式，包括t动机和双t动机之间的配对。接下来，PI将使用这些动机结构来证明变形的多个zeta值的空间是通过某些具有精确算术意义的特殊函数作为tau代数生成的。这部分项目的最终目标是产生一个猜想，描述使用这种tau代数结构描述了多少个zeta值关系。该项目还将包含一个重要的分支，它将这些变形的多个zeta值作为生成元素，在Taelman的单元模块的不可分割的扩展中实现。这部分项目的最终结果将是一个定理，该定理精确地说明我们必须与Taelman的单元模块相邻哪些元素，以确保它包含特定的变形多重zeta值。该项目由数学科学部的代数和数论项目和促进竞争研究的既定项目（EPSCoR）共同资助。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Multiple zeta values are real numbers which are defined by certain infinite sums of fractions. Their exact values and the relationships between them have puzzled mathematicians since the time of Euler. In particular, mathematicians seek to understand when it is possible to add a finite number of multiple zeta values together to get another multiple zeta value. While mathematicians have infinitely many examples of such relationships, proving that these constitute all such relationships remains a distant goal. This project will study near-relatives of multiple zeta values in an alternative setting where there is a reasonable hope of proving such results about the set of all relationships between them. The end goal of the project is to develop a new method of producing families of relationships between multiple zeta values and to classify how many relationships such methods produce. This project will also involve outreach to nearby HBCUs to encourage underrepresented groups to participate in graduate level mathematics and STEM programs.More specifically, the main goal of this project is to lay the groundwork for proving the algebraic independence of certain sets of multiple zeta values defined over global function fields. The first step is to develop a method for generating relations between function field multiple zeta values which are defined over the function field. The PI will do this using his recently developed formulas involving a pairing between t-motives and the dual t-motives. Next, the PI will use these motivic constructions to show that spaces of deformed multiple zeta values are generated as a tau-algebra by certain special functions which have precise arithmetic meaning. The end goal of this part of the project is to produce a conjecture describing how many multiple zeta value relations are described using this tau-algebra structure. This project will also contain a significant offshoot which realizes these deformed multiple zeta values as generating elements in inseparable extensions of Taelman's unit module. The end result of this part of the project will be a theorem which states exactly which elements we must adjoin to Taelman's unit module in order to ensure that it contains specific deformed multiple zeta values.This project is jointly funded by the Algebra and Number Theory program in the Division of Mathematical Sciences and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.