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

这个项目涉及数论和代数几何的边界研究，并扩展到其他几个数学分支。特别是，它涉及多项式方程及其解的工作。与一组多项式方程的解相关的是一个数学对象，即它们的zeta函数。这项研究将深入调查关于这些Zeta函数的猜想，这些猜想是由Alexander Beilinson，Don Zagier等人在30-40年前提出的。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支持研究生在这一研究领域的培训。任何具有整系数的多项式方程组都会产生一个复变量的函数，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，Zeta函数Z(S)在S积分值处的特殊值应表示为周期，即某些特定类型的积分。PI将研究Zeta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Euler-经典的多对数函数，它推广了对数函数。PI使用的技术来自于数学和数学物理的几个部分，一方面是动机理论，另一方面是代数K理论，另一方面是簇簇及其量子化。令人惊讶的是，多对数和集群变种之间存在着深刻的联系。它的应用将远远超出代数几何和数论，例如在理论物理中对散射幅度的研究。他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，包括研究经典和量子多对数、L函数的特殊值、模伽罗瓦群、量子霍奇场理论、团结构和局部系统的模空间的量子化。A)利用簇簇与多对数之间的关系，证明了至少当S=5时，Zagier关于DedekindZeta函数的特定值的猜想，并将基元上同调与多对数基序复合体的上同调联系起来。B)发展量子多重多对数理论，给出穿孔投影线的动机基本群的素数完备化的周期的量子形变。C)综合处理与曲面上G-局部系的模空间相关的各种模的簇结构。将它应用于这些模空间的量子化、量子群的表示理论、镜像对称性和数学物理。D)发展量子霍奇场理论。它的树级给出了Hodge理论的Feynman积分方法。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.