Torsion points on curves of the form $y^n = x^d + 1$ 

The goal of this talk is to classify torsion points on the curve $y^n = x^d + 1$ over the complex numbers, where $n$, $d$ are at least 2 and are coprime. We will motivate the central ideas of the proof by studying the proof of the result by Poonen and Stoll that the only torsion points on a generic hyperelliptic curve are the 2-torsion Weierstrass points. For our case, we will replace the "big geometric monodromy" of their argument with "big Galois action on the torsion of the Jacobian." As a corollary, we will extend the Poonen-Stoll result to superelliptic curves. If time permits, we will explain connections with Jacobi sums, cyclotomic units, Vandiver's conjecture, and Anderson-Ihara theory.