Graded Lie Algebras and Families of Algebraic Curves

In recent work with Jef Laga, we adapt a construction of Slodowy to build families of algebraic curves in graded Lie algebras (this generalizes earlier work of Thorne). This required an understanding of nilpotent orbits in Vinberg representations, and it raised some interesting questions about these orbits that we were able to answer. Our motivation comes from proofs in arithmetic statistics in which orbits in certain representations are used to parametrize rational points on curves. In this talk, Beth Romano gives an introduction to these ideas via examples.