Title: Categorical resolutions of curves and Bridgeland stability

Abstract: Categorical resolutions of singularities are a replacement of resolution of singularities within the realm of triangulated categories. They allow the study of the derived category of a singular variety via a triangulated category that behaves like the derived category of a smooth variety.
In this talk we will discuss an explicit construction of a categorical resolution of singularities for a singular curve, following Kuznetsov and Lunts. We will then produce Bridgeland stability conditions on the categorical resolution, which interpolate between slope-semistability on the singular curve and on its normalization. Finally, we will describe the resulting good moduli spaces of semistable objects, and relate them to classical constructions.