Since Bridgeland’s foundational work, stability conditions on triangulated categories have become a central tool in algebraic geometry, with important applications to moduli spaces, wall-crossing, and derived categories. However, constructing Bridgeland stability conditions on the derived category of a smooth projective variety has remained a difficult problem for more than two decades. Until recently, the main general constructions were largely confined to varieties of dimension at most three. In a recent preprint, Chunyi Li proves the existence of Bridgeland stability conditions on the bounded derived category of any smooth projective variety over C.

In this talk, I will first introduce the definition of Bridgeland stability conditions and discuss some basic examples. I will then review the main ideas behind their construction, with particular emphasis on Li’s recent approach. If time permits, I will also discuss ongoing work aimed at relating Bridgeland stability conditions to more classical notions of stability for coherent sheaves, especially Gieseker stability.