多重zeta值、多重t-zeta值和多重T-zeta值相关问题的研究

近三十年来，由于在量子场论、纽结理论、数论、代数几何等学科中的应用，对多重zeta值相关问题的研究已成为了数学和物理学科中非常活跃的一个领域。近些年，该领域已取得了许多有趣且深刻的结果，但仍然有许多公开的问题和猜想待解决。我们将研究多重zeta值和它们的两类奇偶变形多重t-zeta值与多重T-zeta值的相关问题，特别是它们的算术和代数性质：研究多重zeta值与多重t-zeta值、多重T-zeta值之间的各种相互蕴含关系，以获得它们之间的一些代数刻画；研究多重t-zeta值与多重T-zeta值的相互表示关系；考虑超几何函数与多重t-zeta值、多重T-zeta值的关系，以获得一些母函数关系；考虑多重t-zeta值、多重T-zeta值的反复积分表达式，来研究它们的对偶性、和公式、退化性等算术代数性质，以更好的了解由多重zeta值、多重t-zeta值和多重T-zeta值所张成的向量空间的结构。

In the past three decades, multiple zeta values and related variants have been studied intensively due to their close relations to other objects in many mathematical and physical fields, such as quantum field theory, knot theory and algebraic geometry, etc. It is discovered recently that feynman integrals in perturbative quantum field theory have close relations with multiple polylogarithm values and the renormalization of divergent multiple zeta values. And results of higher-order calculations in quantum electrodynamics and quantum chromodynamics can be simplified by using the shuffle algebraic structure of partial sums of multiple zeta value series. These indicate a deep connection between the theory of motives and the perturbative quantum field theory. This project will provide valuable collaboration opportunities between mathematicians, physicists and computer scientists who work on problems benefiting several diverse fields of sciences at the same time. In recent years, there have been many interesting and profound results in this field, but there are still many open problems and conjectures to be solved. We will systematic study the multiple zeta values and their two odd-even variants, multiple t-zeta values and multiple T-zeta values, especially their arithmetic and algebraic properties: algebra framework, generating function, duality, sum formula, reduction etc. by considering the methods of hypergeometric series, iterated integrals, well-known algebraic relations etc.