Over the course of the last 15 years or so, Minhyong Kim has developed a framework for making effective use of the fundamental group to bound sets of solutions to hyperbolic equations; his method opens a new avenue in the quest for an effective version of the Mordell conjecture. The problem of realising the potential effectivity of his methods has evolved into a wide-ranging, multi-faceted program. Some of its facets are being presented in the talk by Steffen Müller of the previous week, and in the talk by Netan Dogra the following week. My talk will focus on an approach to the case of the unit equation via "motivic" methods. Using these methods we are able to describe an algorithm; its output upon halting is provably the set of integral points, while its halting depends on conjectures.