Beyond geometric invariant theory

Geometric invariant theory is an essential tool for constructing moduli spaces in algebraic geometry, but there are many moduli problems which do not neatly fit into this framework. Recently a theory has emerged which treats the results and structures of geometric invariant theory in a broader context. The theory of Theta-stability applies directly to moduli problems without the need to approximate a moduli problem as an orbit space for a reductive group on a quasi-projective scheme. I will discuss two of the main theorems of geometric invariant theory from this intrinsic perspective – one giving necessary and sufficient conditions for the existence of moduli spaces (joint with Jarod Alper and Jochen Heinloth), and the other giving necessary and sufficient conditions for the existence and uniqueness of  “maximally destabilizing filtrations.” I will discuss applications, such as constructions of moduli spaces for objects in abelian categories, and a general approach to studying the birational geometry of moduli spaces.