(Oxford University)

Manifolds with corners are similar to manifolds, yet are locally modelled on subsets $[0,\infty)^k \times R^{n-k}$. I will discuss some of the theory of these objects, as well as introducing $C^\infty$-rings. This will explain the background to my current research in $C^\infty$-Algebraic Geometry. Time permitting, I will briefly discuss my current research on $C^\infty$-schemes with corners and motivation of this research.

Hyperbolic groups were introduced by Gromov and generalize the fundamental groups of closed hyperbolic manifolds. Since a closed hyperbolic manifold is aspherical, it is a classifying space for its fundamental group, and a hyperbolic group will also admit a compact classifying space in the torsion-free case. After an introduction to this and other topological finiteness properties of hyperbolic groups and their subgroups, we will meet a construction of R. Kropholler, building on work of Brady and Lodha. The construction gives an infinite family of hyperbolic groups with finitely-presented subgroups which are non-hyperbolic by virtue of their finiteness properties. We conclude with progress towards determining minimal examples of the "sizeable" graphs which are needed as input to the construction.

The study of 3-manifolds is founded on the strong connection between algebra and topology in dimension three. In particular, the sine qua non of much of the theory is the Loop Theorem, stating that for any embedding of a surface into a 3-manifold, a failure to be injective on the fundamental group is realised by some genuine embedding of a disc. I will discuss this theorem and give a proof of it.