Rebecca Waldecker
*Martin-Luther-Universität Halle-Wittenberg
***Fixity and subgroup structure
**

In this talk we will look at questions about permutation groups that involve the so-called fixity. We will analyse how small fixity restricts the subgroup structure and how this leads to classification results.

Maximilien Gadouleau
*University of Durham
***Universal sequential boolean automata networks
**

A Boolean Automata Network (BAN) is a set of m automata, each with a boolean variable (its state) which evolves over time according to a deterministic function. The BAN is sequential if at each time step only one automaton updates its state. If at some time step, the states of the first n automata is a function of their initial state (a transformation of \({0,1}^n\)), we say the network computes that transformation. We are then interested in finding universal sequential BANs, which can compute any possible transformation of \({0,1}^n\). We give explicit constructions of such BANs and derive bounds on their minimum space and minimum time. We are then interested in computing all transformations consecutively; again we provide bounds and constructions for BANs which can do so.

Ellen Henke
*University of Copenhagen
***Fusion and Cohomology
**

This talk is about a shared paper with Dave Benson and Jesper Grodal. We prove an elementary group theoretical result which has implications for mod p cohomology and higher chromatic cohomology theories. More precisely, we prove the following result: Suppose we are given a finite group G, an odd prime p, and a subgroup H of G containing a Sylow p-subgroup of G. Then H controls fusion in G if and only if it controls fusion of elementary abelian subgroups. The analogous result is true for p=2 if one considers abelian subgroups of exponent at most 4 instead of elementary abelian subgroups. The proof is more easily carried out in the category of fusion systems than in the category of groups.

Peter Cameron
*University of St Andrews
***Regular polytopes
**

This is joint work with Maria Elisa Fernandes (Aviero), Dimitri Leemans (Auckland) and Mark Mixer (Boston). An abstract polytope is a combinatorial object; it is regular if it has the maximum amount of symmetry (the automorphism group acts transitively on maximal flags). Fernandes, Leemans and Mixer have found some interesting results on regular polytopes whose automorphism group is a symmetric group, and a conjecture on the case of the alternating group. In an attempt to understand these better, we have given an upper bound on the rank of a regular polytope whose group is a transitive subgroup of the symmetric group. The talk will be largely expository.

Jarek Kedra
*University of Aberdeen
***The conjugation invariant geometry of cyclic subgroups
**

I am considering a group \(G\) equipped with the word norm associated with a generating set consisting of all conjugates of finitely many elements. This is a maximal norm among all conjugation invariant norms. I am interested in the growth rate of the sequence \(|g^n|\) of norms of powers of an element \(g\) of \(G\). I will prove that this sequence is either bounded or grows linearly for many classes of groups. I don't know an example of a finitely presented group which has an element of intermediate growth.