There is a growing program to classify symplectic singularities and their resolutions, generalising the Springer resolution of the nilpotent cone, which famously encodes (via quantization) the representation theory of semisimple Lie algebras. For example, Kaledin conjectured these to be conical (and Namikawa and　Odaka proved this under some hypotheses), and Namikawa proved that there are finitely many for each dimension and maximum weight of generator. However, there still seem to be a huge number, so attempts to classify rely on reducing to the most fundamental examples, from which all others can be constructed.

Recently Namikawa produced new examples of Hamiltonian reductions of vector spaces by tori (singular toric hyperkaehler varieties) with trivial local fundamental group. I will explain how to generalise his result to quotients by nonabelian groups. This allows one to construct such examples which are additionally Q-factorial. These can be considered as the fundamental building blocks of conical symplectic singularities, which cannot be resolved or simplified.

Given a general symplectic singularity, one also wishes to understand all its crepant (partial) resolutions. This is closely related to the study of the Cox ring of the resolution. I will explain recent tools to compute this. Finally, I will explain a conjectural program to use Cox rings of deformations to construct all symplectic singularities from the Q-factorial terminal ones via Hamiltonian reduction by tori.