Nonabelian Hodge theory and cohomological DT theory

I'll review the two subjects in the title, with a view to using DT theory to extend nonabelian Hodge theory and the P=W conjecture from the usual setting of smooth moduli varieties to that of singular moduli stacks. This is achieved via a purity result, which may be formulated as the statement that the BBDG decomposition theorem holds for the exceptional direct image of the constant sheaf from the stack of objects in a (reasonable/geometric) 2-Calabi-Yau category to its good moduli space. This in turn follows from a formality result, work of Alper, Hall and Rydh on ├ętale neighbourhoods of good moduli stacks, a thorough understanding of the DT theory of preprojective algebras, and dimensional reduction - I'll try to explain at least the last two of these.