RUI: Multiple Polylogarithms, Multiple Zeta Functions and Related Topics

赵博士将研究算术、代数、几何和数论中的一些问题。他的主要重点是研究的算术，几何和分析性质的多重polyvalues和多重zeta函数这是推广的经典polyvalues和黎曼zeta函数，分别。近年来，这些对象及其各种推广在数学和物理学的许多领域都有突出的表现。理论的特殊价值，特别是，提供并将继续提供答案的重要和深远的问题，如那些在代数几何涉及动机超过一些领域。赵博士将利用Deligned和Goncharov的motivic基本群理论，郭，Kreimer及其合作者开发的Hopf代数技术和Rota-Baxter算子，霍夫曼研究的（准）shuffle代数，以及计算机辅助计算来研究这些特殊值的精细结构。数论是有人类历史记载以来数学的基础之一，算术代数几何是现代数学中最新和最活跃的领域之一，它研究多元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，既丰富了代数几何，又使数论的研究发生了革命性的变化。拟议的研究考虑的问题，涉及对象，深刻反映了一些基本信息领域的代数数，这些对象的定义。这些问题起源于哥德巴赫、欧拉和高斯的工作，几代数学家继续发明新的技术来试图解开它们的谜团。该项目的许多部分通过高级课程作品和夏季研究计划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高年级本科生学习数学，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.