Polylogarithms, Motives, L-Functions, and Quantum Geometry of Moduli Spaces

该项目涉及数论和代数几何边界的研究，并扩展到数学的其他几个分支。它特别关注多项式方程及其解的工作。与一组多项式方程的解相关联的是一个数学对象，它们的ζ函数。这项研究将深入调查亚历山大·贝林森（Alexander Beilinson）、唐·扎吉尔（Don Zagier）等人在30-40年前提出的关于这些ζ函数的一系列猜想。这些研究也对理论物理学产生了影响。该项目的另一个方面是，它将支持这一研究领域的研究生培训。任何具有整系数的多项式方程组都会产生一个只有一个复变量的函数，即Zeta函数Z(s)。它以一种非常神秘的方式编码了解空间中最重要的特征。特别地，函数Z(s)在s的积分值处的特殊值应该表示为周期，即某种特定类型的积分。PI将研究函数的特殊值，将它们与非常经典的数学对象联系起来，追溯到欧拉经典的多对数函数，它推广了对数函数。PI使用的技术来自数学和数学物理的几个部分，例如动机理论，一方面是代数k理论，另一方面是聚类变化及其量化。多对数和聚类之间的深层联系的存在是令人惊讶的。它将有许多远远超出代数几何和数论的应用，例如在理论物理中散射振幅的研究。PI将继续研究各种模空间的量子几何及其在代数几何、数学物理和数论中的应用，包括经典和量子多对数的研究、l函数的特殊值、动力伽罗瓦群、量子霍奇场论、局部系统模空间的簇结构和量子化。本课题的主要重点是：a)利用聚类变异与多对数之间的关系，证明Zagier关于Dedekind zeta函数的特殊值的猜想，至少对于s=5，并将动机上同调与多对数动机复调的动机上同调联系起来。b)发展了量子多重多对数理论，给出了刺穿射影线的动力基群的幂等补全周期的量子变形。c)综合处理曲面上g局部系统的模空间相关的各种模的簇结构。将此应用于这些模空间的量子化、量子群的表示理论、镜像对称和数学物理。d)发展量子霍奇场论。它的树形层次给出了霍奇理论的费曼积分方法。将其作为陈-西蒙斯理论的霍奇理论类比推导出来。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。

This project concerns research at the boundary of Number Theory and Algebraic Geometry and extends to several other branches of mathematics. In particular it concerns work on polynomial equations and their solutions. Associated to the solutions of a collection of polynomial equations is a mathematical object, their zeta function. This research will investigate a deep collection of conjectures concerning these zeta functions, made by Alexander Beilinson, Don Zagier and others 30-40 years ago. These investigations also have consequences for theoretical Physics. Another aspect of the project is that it will support the training of graduate students in this area of research.Any system of polynomial equations with integral coefficients gives rise to a function of one complex variable, the Zeta Function Z(s). It encodes in a very mysterious way the most important characteristics of the space of solutions. In particular, the special values of the zeta function Z(s) at the integral values of s should be expressed as periods, that is integrals of certain specific type. The PI is going to study the special values of the zeta functions, relating them to the very classical mathematical objects, going back to Euler - the classical polylogarithm functions, which generalize the logarithm function. The technique the PI is using came from several parts of Mathematics and Mathematical Physics, such as the theory of motives, algebraic K-theory on one hand, and cluster varieties and their quantization on the other. The very existence of deep connections between polylogarithms and cluster varieties is surprising. It will have many applications far beyond Algebraic Geometry and Number Theory, e.g. in the investigation of Scattering Amplitudes in Theoretical Physics. The PI will continue his work on quantum geometry of various moduli spaces and their applications in Algebraic Geometry, Mathematical Physics and Number Theory, including the study of classical and quantum polylogarithms, special values of L-functions, motivic Galois groups, Quantum Hodge Field Theory, cluster structure and quantization of moduli spaces of local systems. The main priorities of the project are the following:a) To use the relationship between cluster varieties and polylogarithms to prove Zagier's conjecture on the special values of the Dedekind zeta function at least for s=5, and relate the motivic cohomology to the cohomology of the polylogarithmic motivic complexes. b) To develop the theory of quantum multiple polylogarithms, providing a quantum deformation of the periods of the prounipotent completion of the motivic fundamental group of the punctured projective line. c) To give a comprehensive treatment of the cluster structure of various moduli related to the moduli spaces of G-local systems on surfaces. Apply this to quantization of these moduli spaces, representation theory of quantum groups, mirror symmetry and Mathematical Physics. d) Develop Quantum Hodge Field Theory. Its tree level gives a Feynman integral approach to Hodge theory. Derive it as a Hodge-theoretic analog of Chern-Simons theory.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.