Georgia Institute of Technology College of Sciences

Brill--Noether theory studies linear systems on an algebraic curve. K3 surfaces have been central in Brill--Noether theory since Lazarfeld's proof of the Brill--Noether theorem, which describes the linear series on a general curve of genus $g$. In this talk, we'll survey some recent results concerning the Brill--Noether theory of curves on K3 surfaces. Specifically, we'll discuss a refined Brill--Noether theory, which aims to understand the linear series on curves with a given Brill--Noether special linear series, and related unstable vector bundles on K3 surfaces with applications to the dimensions of components of Giesker--Petri loci in $\mathcal{M}_g$, parameterizing curves admitting a linear series with non-injective Petri map.