Title: Semi-orthogonal decompositions of moduli spaces

Abstract: I will discuss strategies for combining categorical and geometric techniques to decompose derived categories into semi-orthogonal indecomposable blocks focussing on two classical moduli spaces. For the moduli space of stable rank 2 vector bundles with a fixed odd determinant on a smooth algebraic curve, jointly with Sebastiá Torres, we construct an SOD with blocks given by symmetric powers of the curve, confirming a conjecture of Narasimhan and Belmans-Galkin-Mukhopadhyay. For the moduli space of stable pointed curves of genus 0, jointly with Ana-Maria Castravet, we construct an exceptional collection invariant under the symmetric group of permutations of marked points, proving a conjecture of Manin and Orlov.