Stability conditions on singular varieties so far have only been constructed for curves and a few types of surfaces. In this talk, we will show a construction on surfaces with ADE singularities, and study the compatibility between stability conditions on the singular surfaces and those on the resolutions.
Moreover, the argument may also be a useful model when studying the case of Fano threefolds, where we naturally focus more on the Kuznetsov components. 

Finally, we will introduce the Kuznetsov-Shinder bridge construction and talk about some possible applications.