Geometric functional equations for polylogarithms

A well-known conjecture of Zagier states that the value of a Dedekind zeta function of a number field at an integer m>1 can be expressed in terms of a single transcendental function, namely the m-th polylogarithm function. This conjecture has been proved for m=2 and m=3, but remains open for all m>3. The main difficulty in proving the conjecture for m>3 comes from the lack of knowledge about the structure of functional equations for higher polylogarithms. The aim of this talk is to report on some recent progress in this direction in the case m=4.