Abstract: In this talk, we will introduce new affine algebraic varieties 
for algebras given by quiver and relations. Each variety contains a 
distinguished element in the form of a monomial algebra. The properties 
and characteristics of this monomial algebra govern those of all other 
algebras in the variety. We will show how amongst other things this gives 
rise to a new way to determine whether an algebra is quasi-hereditary. 
This is a report on joint work both with Ed Green and with Ed Green and 
Lutz Hille.

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.

 

In this talk two different methods for constructing complete steady and expanding Ricci solitons of cohomogeneity one will be discussed. The first is based on an estimate on the growth of the soliton potential and holds for large classes of cohomogeneity one manifolds. The second approach is specific to the two summands case and uses a Lyapunov function. This method also carries over to the Einstein case and as an application, a simplified construction of B\"ohm's Einstein metrics of positive scalar curvature on spheres will be explained.

In this talk we describe an approach to approximate the truncated singular value decomposition of a large matrix by first decomposing the matrix into a sum of Kronecker products. Our approach can be used to more efficiently approximate a large number of singular values and vectors than other well known schemes, such as iterative algorithms based on the Golub-Kahan bidiagonalization or randomized matrix algorithms. We provide theoretical results and numerical experiments to demonstrate accuracy of our approximation, and show how the approximation can be used to solve large scale ill-posed inverse problems, either as an approximate filtering method, or as a preconditioner to accelerate iterative algorithms.