Brill–Noether loci, which parameterize algebraic curves of fixed genus
with unexpected linear systems, are important subvarieties of the
moduli space of curves. The behavior of linear systems on general curves in these loci is still largely mysterious, though recent advances have been made in the case of plane curves and curves of fixed gonality. In this direction, I'll discuss the Maximal Brill-Noether Loci Conjecture and its resolution using new techniques from the theory of Lazarsfeld-Mukai bundles on Brill-Noether special K3 surfaces.