Birational invariants of algebraic groups and algebraic tori with finite group actions

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A classical but notoriously difficult problem in algebraic geometry is to classify algebraic varieties up to birational isomorphism - a natural equivalence relation on the set of algebraic varieties. Rational algebraic varieties form a distinguished class under this equivalence relation. The simplest known rational varieties are the linear varieties and the algebraic tori. There are natural actions of finite groups on linear varieties and algebraic tori coming from representations of the groups. One may ask when the orbit space of a linear variety or algebraic tori under a finite group action is rational. This question was first posed by Emmy Noether while she was doing work on the inverse Galois problem. One may also compare an algebraic torus with a finite group action to the associated linear variety with a finite group action by asking when the two are birationally isomorphic with a birational isomorphism which is equivariant with respect to the finite group action. This question of equivariant birational linearisation is related to the question of finding conjugacy classes of finite subgroups in the classical Cremona group - the group of birational isomorphisms of projective space. It is also related to the classical problem of determining whether an algebraic group is Cayley - or equivariantly birationally isomorphic to its Lie algebra - a problem first studied by Cayley. I previously did joint work on this problem with Vladimir Popov and Zinovy Reichstein. Among other things, we determined the set of simple algebraic groups over an algebraically closed field which are Cayley. I study analogues and generalisations of the rationality problem for algebraic tori under finite group actions and also the equivariant birational linearisation problem.