Abstract: We prove that the weight 4 Beilinson's regulator map can be expressed via the classical n-logarithms, $n \leq 4$.
This plus Borel's theorem implies Zagier's conjecture, relating the value of the Dedekind zeta functions at $s=4$ and the classical tetralogarithm. Another application is to the values of L-functions of elliptic curves over $Q$ at $s=4$.
One of the new tools is a connection between cluster varieties and polylogarithms, generalising our work with V. Fock relating cluster varieties and the dilogarithm.
It allows to describe explicitly the weight less or equal to four quotient of the motivic Tate Galois Lie algebra related to the punctured projective line.
This is joint work with Daniil Rudenko (U Chicago).