Abstract: This is joint work with Ragnar-Olaf Buchweitz and Colin Ingalls. 
The classical McKay correspondence relates the geometry of so-called 
Kleinian surface singularities with the representation theory of finite 
subgroups of SL(2,C). M. Auslander observed an algebraic version of this 
correspondence: let G be a finite subgroup of SL(2,K) for a field K whose
characteristic does not divide the order of G. The group acts linearly on 
the polynomial ring S=K[x,y] and then the so-called skew group algebra
A=G*S can be seen as an incarnation of the correspondence. In particular
A is isomorphic to the endomorphism ring of S over the corresponding 
Kleinian surface singularity.
Our goal is to establish an analogous result when G in GL(n,K) is a finite 
subgroup generated by reflections, assuming that the characteristic
of K does not divide the order of the group. Therefore we will consider a 
quotient of the skew group ring A=S*G, where S is the polynomial ring in n 
variables. We show that our construction yelds a generalization of 
Auslander's result, and moreover, a noncommutative resolution of the 
discriminant of the reflection group G.

Abstract: Joint work with Carlson, Grodal, Nakano. In this talk we will
present some recent results on an 'important' class of modular 
representations for an 'important' class of finite groups. For the 
convenience of the audience, we'll briefly review the notion of an 
endotrivial module and present the main results pertaining endotrivial 
modules and finite reductive groups which we use in our ongoing work.

I shall describe recent work with Srikanth Iyengar, Henning 
Krause and Julia Pevtsova on the representation theory and cohomology
of finite group schemes and finite supergroup schemes. Particular emphasis 
will be placed on the role of generic points, detection of projectivity
for modules, and detection modulo nilpotents for cohomology.