**Speaker:** Tom Coleman (St Andrews)

**Title:** Automorphisms of MB-homogeneous graphs

**Abstract: **A *bimorphism* of a graph (or more generally, a first-order structure ) is a bijective homomorphism from to itself. If is infinite, a bimorphism may not necessarily be an automorphism of . It follows that the collection of all bimorphisms of a graph is a submonoid of the full symmetric group on the vertices of ; this is then a natural example of a *permutation monoid*. Generalising the model-theoretic idea of a homogeneous structure, a graph is *MB-homogeneous *if every injective monomorphism between finite induced subgraphs of extends to a bimorphism of .

This talk will explore MB-homogeneous graphs in more detail, attempting to keep the associated model theory minimal. We will show that MB-homogeneous graphs give examples of *oligomorphic permutation monoids, *give many examples of MB-homogeneous graphs with a surprising property, and show that any finite group H arises as the automorphism group of an MB-homogeneous graph.

This talk is based on joint work with David Evans (Imperial) and Bob Gray (UEA).

**Speaker:** Alan Logan (Glasgow)

**Title:** One relator groups with torsion

**Abstract: **One relator groups were first studied by Magnus in the 1930s. One relator groups which contain torsion elements (non-trivial elements of finite order) are hyperbolic. In this talk we will explore how (and, I claim, why) one relator groups with torsion bridge the gulf in difficulties between dealing just with torsion free hyperbolic groups and dealing with all hyperbolic groups.

**Speaker:** Jenny August

**Title: **Braid relations, hyperplane arrangements and the homological minimal model program

**Abstract: **Given a Coxeter group *W*, there is a natural surjection from *A* to *W* where *A* is the associated Artin group. The kernel of this map, known as the pure braid group, is generally difficult to write down in terms of generators and relations. However, in the 1970’s Deligne showed it could be described using the topology of an associated hyperplane arrangement.

In this talk, I will describe this result, and more generally his result for simplicial hyperplane arrangements before then going on to describe an application to algebraic geometry. In particular, we show certain functors arising in this area satisfy higher length braid relations.