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.
