**Speaker:** Valentina Grazian (Aberdeen)

**Title:** Fusion and pearls: what is known

**Abstract: **In finite group theory, the word *fusion* refers to the study of conjugacy maps between subgroups of a group. The modern way to solve problems involving fusion is via the theory of fusion systems. A saturated fusion system on a *p*-group *S* is a category whose objects are the subgroups of *S* and whose morphisms are the monomorphisms between subgroups which satisfy certain axioms. The first part of this talk is about the state of the art concerning the classification of simple fusion systems.

To classify saturated fusion systems, we first need to determine the so-called *essential *subgroups of *S*. These are self-centralizing subgroups of *S* whose automorphism group has a restricted structure. We call *pearls* the essential subgroups of *S* that are either elementary abelian of order *p*^{2} or non-abelian of order *p*^{3} (and exponent *p* whenever *p* is odd). In the second part of this talk we present new results about fusion systems involving pearls and we explain how such results contribute to the classification of simple fusion systems.

**Speaker:** Rosemary Bailey (St Andrews)

**Title:** Some applications of finite group theory in the design of experiments

**Abstract: **Group theory is used in (at least) two different ways in the design

of experiments. The first is in randomization, the process by which an initial

design is turned into the actual layout for the experiment by applying a

permutation of the experimental units, chosen at random from a certain

group of permutations. Which group? What properties should it have? The second

is in design construction. The set of treatments is identified with a finite

Abelian group, and the blocks are all translates of one or more initial blocks.

The characters of this group form its dual group: they are the eigenvectors

of the matrix that we need to consider to see how good the proposed design is.

**Speaker:** JB Gramain (Aberdeen)

**Title:** Basic sets and perfect isometries

**Abstract: **Basic sets can be a powerful tool in modular representation theory of finite groups, sometimes helping in constructing Brauer characters or in reducing the problem of determining decomposition matrices.

In this talk, I will present some fairly old results on basic sets for the symmetric and alternating groups, and show how the methods and results can be generalised to the double Schur covers of these groups when the characteristic is odd. In all these groups, we exhibit basic sets by using perfect isometries between blocks of complex characters.

This is joint work with Olivier Brunat (Université Paris 7).