RUI: Multiple Polylogarithms, Multiple Zeta Functions and Related Topics

赵博士将研究算术、代数、几何和数论方面的一些问题。他的主要研究方向是多重多对数和多重zeta函数的算术、几何和解析性质，它们分别是经典多对数和黎曼zeta函数的推广。近年来，这些对象及其各种推广在数学和物理的许多领域中占有突出地位。特别是关于它们的特殊值的理论，已经并将继续为一些重要而深远的问题提供答案，例如代数几何中涉及数域动机的问题。Zhao博士将利用deldesigned和Goncharov的动机基本群理论，由Guo， Kreimer及其合作者开发的Hopf代数技术和Rota-Baxter算子，Hoffman研究的（拟）洗牌代数，以及计算机辅助计算来研究这些特殊值的精细结构。自从有记载的人类历史开始以来，数论就是数学的基础之一，现在它作为许多应用的基础，包括密码学和编码理论。算术代数几何是现代数学中最新和最活跃的领域之一，它研究多变量多项式方程组解的几何性质的算术性质。它在数论中的应用不仅丰富了代数几何，而且使数论研究发生了革命性的变化。提出的研究考虑了涉及对象的问题，这些对象深刻地反映了这些对象所定义的代数数域的一些基本信息。哥德巴赫（Goldbach）、欧拉（Euler）和高斯（Gauss）的著作中都有这样的问题，一代又一代的数学家不断发明新技术，试图解开这些谜团。该项目的许多部分通过高级课程和暑期研究项目为本科生提供了重要的研究机会。赵博士计划利用这些机会吸引更多的高级本科生学习数学，让他们参与数学研究的各个阶段，从最初的计算到最终在各种专业会议上展示他们的结果。

Dr. Zhao will work on a number of problems in arithmetic algebraic geometry and number theory. His main focus is on the study of arithmetic, geometric and analytic properties of the multiple polylogarithms and the multiple zeta functions which are generalizations of the classical polylogarithms and the Riemann zeta function, respectively. In recent years, these objects and their various generalizations have appeared prominently in a lot of areas of mathematics and physics. The theory of their special values, in particular, has provided and will continue to provide answers to important and far reaching problems such as those in algebraic geometry involving motives over number fields. Dr. Zhao will utilize the theory of motivic fundamental groups of Deligned and Goncharov, Hopf algebra techniques and Rota-Baxter operators developed by Guo, Kreimer and their collaborators, (quasi-)shuffle algebras studied by Hoffman, and computer-aided computation to investigate the fine structures of these special values.Number theory is one of the foundations of mathematics since the beginning of recorded human history, and it serves nowadays as the basis for many applications, including cryptography and coding theory.Arithmetic algebraic geometry, one of the newest and most active fields of modern mathematics, studies the arithmetic nature of geometric properties of solutions to systems of polynomial equations in several variables. Its application in number theory has both enriched algebraic geometry and revolutionized the study of number theory. The proposed research considers questions involving objects that deeply reflect some fundamental information about fields of algebraic numbers over which these objects are defined. Such questions have their genesis in the work of Goldbach, Euler and Gauss, and mathematicians in generations continue to invent new techniques to try to solve their mysteries. Many parts of the project offer significant research opportunities for undergraduate students through both advanced course works and the summer research programs. Dr. Zhao plans to utilize these opportunities to attract more advanced undergraduate students to study math by involving them in mathematical research in all the stages, from the initial computation to the final presentation of their results in various professional meetings.