A Lie-theoretic trichotomy in Diophantine geometry and arithmetic dynamics

How can the finite/infinite dichotomy of the Killing–Cartan classification of simple Lie groups & algebras appear in number theory? I will explain how this Lie-theoretic dichotomy is realized in the finiteness or infinitude of positive integer solutions to certain Diophantine equations, and explore some of its implications for classical questions studied by Gauss, Mordell, Coxeter, Conway, and Schinzel in combinatorics and number theory. I will then switch gears to the arithmetic dynamics of cluster Donaldson–Thomas transformations, which refines the Diophantine realization of the finite/infinite dichotomy into a finite/affine/indefinite trichotomy that matches the Kac–Moody classification of infinite-dimensional Lie algebras.