The notion of Bridgeland stability conditions is the derived analogue of the slope stability of coherent sheaves.
Like the ample cone of a variety,
the space of Bridgeland stability conditions reflects properties of a variety.
The noncommutative minimal model program (NCMMP), proposed by Halpern-Leistner, is an analogue of MMP in derived categories. The NCMMP aims to divide the derived category into smaller components through semiorthogonal decomposition by using the space of Bridgeland stability conditions.
In this talk, I will show the relation between the NCMMP and the quantum cohomology, and I will discuss the blowup of a surface at a point.