For a general algebraic stack X, we will present combinatorial structures underlying the connected components of the stack of filtrations of X. This allows us to define analogues of the Hall algebra in different flavors, except that we do not get algebras but a more general kind of structure. Classically these Hall algebras were only defined when X parametrizes objects in an abelian category, while our construction is general. We will then discuss applications of the theory. 

In the motivic setting, we define a notion of Euler characteristic for a stack and, in the (-1)-shifted symplectic case, we give an intrinsic definition of Donaldson–Thomas invariants. The construction relies on a no-pole theorem. The invariants depend on the choice of a so-called stability measure. The space of such measures is a unipotent algebraic group that governs how invariants change under wall-crossing.

In the cohomological setting, we get an explicit form of the decomposition theorem for the map from the stack to its good moduli space, in the smooth, 0-symplectic, and (-1)-symplectic case, assuming tangent space representations at closed points are orthogonally symmetric.

This is joint work over different projects with Chenjing Bu, Ben Davison, Daniel Halpern-Leistner, Tasuki Kinjo and Tudor Pădurariu.