In recent work with Jef Laga, we give a way to build families of algebraic curves in graded Lie algebras. Our motivation comes from proofs in arithmetic statistics in which orbits in certain representations are used to parametrize rational points on curves. This generalizes earlier work of Jack Thorne. In the talk, I won’t assume prior knowledge of graded Lie algebras or arithmetic statistics. I’ll give an introduction to these areas via examples.