Abstract:
This talk will connect algebraic geometry and representation theory and will be self-contained. We will discuss projective curves over a fixed algebraically closed field $k$, and then define the sheaf of relative differentials on a projective curve over $k$. The global sections of this sheaf form a vector space called holomorphic differentials. Letting a finite group $G$ act on the projective curve, the space becomes a module over the group algebra $kG$, and hence we can ask about its group representation. Finally, we will discuss some geometric methods used in finding this spaces' representation theory.