Given a simply laced Dynkin diagram, one can use Vinberg theory of graded Lie algebras to construct a family algebraic curves. In the case when the diagram is of type E7 or E8, Jack Thorne and I have used the relationship between these families of curves and their associated Vinberg representations to gain information about integral points on the curves. In my talk, I'll focus on the role Lie theory plays in the construction of the curves and in our proofs. Speaker(s): Beth Romano (University of Cambridge)