The classical theory of multiple polylogarithms found a wealth of interesting relations between interesting transcendental numbers.
It also accounted, at least conjecturally, for the transcendental part of certain special values of L-functions. This led, through the theory of regulators, to the search for associated elements of K-groups. Goncharov’s theory places these K-groups inside a certain graded Hopf algebra of “framed mixed Tate motives”. The classical regulators are then related to certain transcendental period maps. The special elements are vastly generalized by Goncharov’s motivic multiple polylogarithms, whose periods are the classical multiple polylogarithms.
Goncharov’s theory led to a series of intricate conjectures about the structure of the mixed Tate Hopf algebra, and has placed this object in the center of a growing area of research. In ongoing joint work with David Corwin, we use one of Goncharov’s more intricate conjectures to make my “Kim-theoretic” algorithm for computing solutions to the unit equation (partly joint with Stefan Wewers) simpler and clearer. As applications, we hope to obtain numerical evidence for Goncharov’s conjecture, as well as for Kim’s nonabelian Shafarevich-Tate conjecture.