The Mullineux problem is a classical topic in the modular representation theory of the symmetric group. The aim is to understand how the sign representation twist the irreducible representations of the symmetric group in positive characteristic. In this talk, we give a new approach for this problem by using the representations of affine Hecke algebras and Ariki-Koike algebras.

The Hochschild cohomology (the HH*) of finite dimensional algebras is a useful tool to help with classifying blocks of group algebras. For example, if k is a field and A, B are two symmetric k-algebras, then a stable equivalence of Morita type between A and B induces an isomorphism between the HH^n of A and B in degree n greater than zero. What is more, the graded ring HH* forms a Gerstenhaber algebra, and in particular the HH^1 forms a Lie algebra over k, whose structure is also preserved under stable equivalence of Morita type.
In this talk I will discuss a range of topics and questions regarding the HH^1 of blocks of finite group algebras. These will include results on blocks with a cyclic or low rank elementary abelian defect group, the use of a powerful computational result to calculate the dimensions of the HH^1 of blocks, calculation of the Lie algebra structure of the HH^1 of blocks of the Mathieu groups, and results on the HH^1 of twisted group algebras.

We present a characterization of the groups possessing a 2-generated Sylow 2-subgroup in terms of their character theory. This talk is based on joint works with G. Navarro, N. Rizo and A. A. Schaeffer Fry.

Picard groups of algebras have extensively been studied in the past, but just recently people started looking at Picard groups of blocks of finite groups. These revealed themselves to be useful tools, for example for dealing with Donovan conjecture, but they’re also interesting in their own right, since they have the structure of a finite group. In this talk we will give an introduction to Picard groups and Picent of blocks, and then present joint work with Michael Livesey on blocks with normal defect groups, providing evidence to a conjecture on basic Morita equivalences.

In a current on-going programme to prove some of the local global counting conjectures a key ingredient requires an understanding of characters in the normalisers of \(d\)-split Levi subgroups. In this talk I will present recent results in this direction for the symplectic groups.