H E A (Eddy) Campbell (Professor)
My overall goal for nearly 3 decades has been to understand the modular analogues of the theorems of classical invariant theory. For example, work of Coxeter, Chevalley, Shephard and Todd, and Serre showed that the rings of invariants of finite non-modular groups are polynomial if and only if the (representation of the) group is generated by (pseudo-)reflections, demonstrating the strong relationship between the geometry of the representation and the algebraic properties of the invariant ring.
The modular analogue of this theorem is still open - the case when the characteristic of the field divides the order of the group. In addition to the difficulties posed by the non-invertibility of the order of the group in the field, the under-lying representation theory is known to be wild.
Advances continue to be made, however: Reiner and colleagues with results giving modular versions of classical theorems concerning co-invariants; Symonds using Castelnuovo-Mumford regularity to bound the top degree occurring in a generating set (and much more); Braun using the canonical module to obtain modular analogues of classical theorems concerning Gorenstein rings of invariants; Wehlau using Roberts isomorphism to understand the invariants of cyclic p-groups in characteristic p; Kemper's methods regarding the Cohen-Macaulay locus; and the work of my colleagues (Chuai, Shank, Wehlau) and myself in understanding modular rings of invariants, particularly vector invariants.