Polylogarithms, Mixed Motives and Special Values of L-Functions

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ABSTRACT Alexander Goncharov Brown University 98 00998 Professor Goncharov will continue his work on classical polylogarithms, and their generalizations, special values of L-functions of algebraic varieties, algebraic K-theory, and motivic Galois groups. In particular the investigator will consider the Hodge theoretic analogs of polylogarithms, as well as classical and Grassmannian polylogarithms. He will study analogs of a conjecture of Zagier to help investigate the Galois group of the category of mixed motives generated by an elliptic curve. Further Professor Goncharov will try to establish the basic properties of higher Arakelov Chow groups This project involves mathematics on the borderlines of Algebra, Number Theory and Geometry in an area known as algebraic K-theory. K-theory represents a powerful method in the modern approach to many mathematical studies. Mathematicians have know for quite a long time that the key to understanding a mathematical construction is in understanding its symmetries. In some sense, K-theory formalizes the study of the algebraic symmetries of mathematical objects. 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.