Finite linear groups generated by reflections arise in many areas of algebra, Lie theory being a prominent example. In the work of Chevalley, Shephard, Todd and Serre, reflection groups are seen to be the groups which have ""good"" rings of invariants when acting on a ring of polynomials.
More recently, reflection groups have been studied in connection with noncommutative rings that are obtained from commutative rings via deformation (or quantisation) construction inspired by quantum mechanics. In particular, this has led to the rich theory of Cherednik algebras.
I am interested in reflection groups, and their quantum analogues, acting on noncommutative rings arising from quantum algebra. Projects might focus on open conjectures in this area, and should be suitable for students with background, and interest, in representation theory and/or quantum groups.