In 2018, Lawrence and Venkatesh gave a new proof of Mordell's conjecture (Faltings' theorem) via a striking argument involving variations of $p$-adic Galois representations. This can be viewed as a $p$-adic period mapping. Their argument essentially decomposes the proof of Mordell's conjecture into an arithmetic and geometric component. In this talk I will explain briefly their argument, and in particular show that the geometric component of their argument is uniform. In particular, we show that the so-called uniform boundedness conjecture of rational points of curves of genus $g \geq 2$ in a twist family is reduced to a statement about uniform finiteness of global Galois representations related to a 1-parameter family of curves. As a concrete application, we show that the geometric part of Lawrence and Venkatesh's argument gives a new proof of a uniform boundedness result for $S$-unit equations, which qualitatively recovers a result of Evertse, which is best possible in light of lower bounds constructed by Erdos, Sarkozy, and Stewart. This is joint work with Brett Nasserden (Waterloo).