Abstract: I will describe how the ADE preprojective algebras appear in
certain Conformal Field Theories, namely SU(2) WZW models, and explain
the generalisation to the SU(3) case, where 'almost CY3' algebras appear.
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.
Oxford Mathematician Ulrike Tillmann FRS has been elected a member of the Council of the Royal Society. The Council consists of between 20 and 24 Fellows and is chaired by the President.
How does the skin develop follicles and eventually sprout hair? Research from a team including Oxford Mathematicians Ruth Baker and Linus Schumacher addresses this question using insights gleaned from organoids, 3D assemblies of cells possessing rudimentary skin structure and function, including the ability to grow hair.
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.
