Abstract: 
Given a Fano manifold M with extremal contractions to Fano manifolds A and B, it is expected that the derived category of M contains two semiorthogonal decompositions, related by the action of the braid group, which refine the semiorthogonal decompositions of the derived categories of A and B. I will discuss strategies for proving this expectation when the Fano manifolds have moduli interpretations. As one application, we construct a semiorthogonal decomposition of the derived category of the moduli space of stable rank 2 vector bundles on a smooth projective curve, as conjectured by Narasimhan and by Belmans, Galkin, and Mukhopadhyay.
