Bristol

I will describe how to build a noncommutative ring which dictates

the process of resolving certain two-dimensional quotient singularities.

Algebraically this corresponds to generalizing the preprojective algebra of

an extended Dynkin quiver to a larger class of geometrically useful

noncommutative rings. I will explain the representation theoretic properties

of these algebras, with motivation from the geometry.

There are many triangulated categories that arise in the study

of group cohomology: the derived, stable or homotopy categories, for

example. In this talk I shall describe the relative cohomological

versions and the relationship with ordinary cohomology. I will explain

what we know (and what we would like to know) about these categories, and

how the representation type of certain subgroups makes a fundamental

difference.

We consider as a starting point a problem raised by Kunen and Tall as to whether

the real continuum can have non-homeomorphic versions in different submodels of

the universe of all sets. Its resolution depends on modest large cardinals.

In general Junqueira and Tall have made a study of such "substructure spaces"

where the topology of a subspace can be different from the usual relative

topology.