**Speaker:** Ellen Henke (University of Aberdeen)

**Title:** Fusion systems

**Abstract: **The theory of saturated fusion systems generalizes important aspects of finite group theory, since each finite group leads to a saturated fusion system which encodes the conjugacy relations between p-subgroups. In particular, the study of saturated fusion systems relates to questions in homotopy theory, the modular representation theory of finite groups, and the group theory surrounding the proof of the classification of finite simple groups. I will give a survey talk on the subject.

**Speaker:** Murray Elder (University of Newcastle, Australia)

**Title:** Solving twisted word equations in PSPACE

**Abstract: **This is joint work with Volker Diekert, Stuttgart. Solving equations in virtually free groups reduces to the problem of solving \emph{twisted equations} in free groups. Building on our previous work with Ciobanu we give a PSPACE algorithm to find all solutions and express them as a formal language of low complexity – EDT0L. I will try to give a sketch of our method and show how twisting makes everything considerably harder (or more fun).

**Speaker:** Peter Cameron (University of St Andrews)

**Title:** Regular semigroups and the existential transversal property

**Abstract: **For the last decade or so, João Araújo and I have been considering the question: what properties of a permutation group *G* on *X* ensure that the transformation semigroup *<G,a>* has some nice property such as regularity for all maps *a:X→X* which are not permutations, or all such maps under some restriction? For *k≤n/2*, the semigroup *<G,a>* is regular for all maps of rank *k* if and only if *G* has the *k-universal transversal property*: given any *k*-set *A* and *k*-partition *P*, there is an element of *G* mapping *A* to a transversal for *P*. We have an almost complete classification of such groups. If we only ask for the property to hold for all maps with prescribed image, we have to consider the *k-existential transversal property*, and things are more complicated. However, we have found a number of results on this.