Invariants for algebraic group actions

The study of rings of invariants has a long and illustrious history (see, for example, Hilbert's fourteenth problem). One of the highlights of the theory in the twentieth century was the proof that rings of invariants for reductive algebraic groups acting on (the coordinate rings of) affine varieties are finitely generated -- this implies that there is a useful notion of quotient variety in this situation. Finite generation fails in general if the group acting is not reductive, but it is often possible to retrieve some aspects of the theory. One example of this is the notion of "separating invariants" for algebraic group actions -- these are finite sets of invariants which can separate as much as the whole ring of invariants, even when the full ring of invariants is not finitely generated. The aim of this project is to study invariants for algebraic group actions in non-reductive cases, starting by building on work of Dufresne and co-authors for invariants of additive group actions.