There are two main approches to motivic (co-)homology appearing in recent years. First, one tries to construct it as cohomology groups of certain complexes with terms being given by explicit generators and relations generalizing Milnor's K-groups. Second, one can construct cohomology of a scheme as cohomology of a complex defined in terms of algebraic cycles thus generalizing the classical defintion of Chow groups. Each has its own advantages and disadvantages. Dr. Jianqiang Zhao's research will continue his work on understanding both approaches by using polylogarithms and their generalizations. Polylogarithms have been around for centuries in one form or another, but only recently has their whole theory become available. They have deep connections with both number theory and modern mathematical physics. This project should shed light on these connections.
This research is in the field of arithmetic algebraic geometry, a subject that combines the techniques of algebraic geometry and number theory. In its original formulation, algebraic geometry treated figures that could be defined in the plane by the simplest equations, namely polynomials. Number theory is the study of numbers that can be expressed in terms of whole numbers, 1, 2, 3... In the second half of last century these two seemingly far apart subjects have produced tremendous impact on each other. The field of arithmetic algebraic geometry now uses techniques from all of modern mathematics.