Deformations and derived categories of quiver moduli

Vector bundles on curves and quiver representations have moduli theories that are very similar. Inspired by this connection, I will recall the canonical 4-term sequence on quiver moduli and its relationship to the Kodaira-Spencer morphism. This sequence can be used to prove a conjecture of Schofield, establish the rigidity of quiver moduli, and show that the universal representation furnishes a fully faithful Fourier-Mukai functor. The stacky perspective of these moduli spaces and Teleman quantization play an important role. If time permits, I will also discuss the Chow rings of quiver moduli. This combines various joint works-in-progress with Ana-Maria Brecan, Hans Franzen, Gianni Petrella, and Markus Reineke.