Non-reductive GIT for graded groups and curve counting

I will start with a short report on recent progress in constructing quotients by actions of non-reductive algebraic groups and extending Mumford's geometric invariant theory to a wide class of non-reductive linear algebraic groups which we call graded groups. I will then explain how certain components of the Hilbert scheme of points on smooth varieties can be described as non-reductive quotients and why this description is especially efficient to study the topology of Hilbert schemes. In particular I will explain how equivariant localisation can be used to develop iterated residue formulae for tautological integrals on geometric subsets of Hilbert schemes and I present new formulae counting curves on surfaces (and more generally hypersurfaces in smooth varieties) with given singularity classes. This talk is based on joint works with Frances Kirwan, Thomas Hawes, Brent Doran and Andras Szenes.