Multiple polylogarithms: analytic continuation, monodromy, and variations of mixed Hodge structure, Contemporary trends in algebraic geometry and algebraic topology

Abstract. In this exposition we shall describe a new way to analytically continue the multiple polylogarithms by using Chen’s theory of iterated path integrals. Then we explicitly determine the good unipotent variations of mixed Hodge-Tate structures (MHS) related to multiple logarithms and some other multiple polylogarithms of lower weights. Following Deligne and Beilinson we define the single-valued real analytic version of the multiple polylogarithms which generalizes the well-known result of Zagier on classical polylogarithms. At the end, motivated by Zagier’s conjecture we pose a problem which relates the special values of multiple Dedekind zeta functions of a number field to the single-valued version of multiple polylogarithms. The main results of this paper with complete proofs will appear elsewhere