09:00 to 10:00 Martin Liebeck The subgroup structure of almost simple groups, I In these two talks I will discuss the subgroup structure of almost simple groups, the emphasis being mainly on maximal subgroups. The first talk will focus on alternating and classical groups, and the second on exceptional groups of Lie type. INI 1 10:00 to 11:00 Eamonn O'Brien Algorithms for matrix groups I Most first-generation algorithms for matrix groups defined over finite fields rely on variations of the Schreier-Sims algorithm, and exploit the action of the group on an set of vectors or subspaces of the underlying vector space. Hence they face serious practical limitations. Over the past 25 years, much progress has been achieved on developing new algorithms to study such groups. Relying on a generalization of Aschbacher's theorem about maximal subgroups of classical groups, they exploit geometry arising from the natural action of the group on its underlying vector space to identify useful homomorphisms. Recursive application of these techniques to image and kernel now essentially allow us to construct in polynomial time the composition factors of the linear group. Using the notion of standard generators, we can realise effective isomorphisms between a final simple group and its "standard copy". In these lectures we will discuss the "composition tree" algorithm which realises these ideas; and the "soluble radical model" which exploits them to answer structural questions about the input group. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Britta Späth Reduction Theorems for global-local Conjectures This talk should give an overview of the reduction theorems proven in the last years. The aim is to prove certain long-standing conjectures relating representations of finite groups to the ones of certain subgroups by reducing them to statements about representations of finite quasi-simple groups and use the classification of finite simple groups. Key to the proof of the reduction theorems is a refinement of Clifford theory. These results have also put forward numerous questions on the representations of finite simple groups. INI 1 12:30 to 14:00 Lunch at Westminster College 14:00 to 15:00 Klaus Lux Basic algorithms in the representation theory of finite groups and algebras The aim of this talk is an introduction to the basic algorithms in computational representation theory. We will start with the fundamental MeatAxe algorithm for proving the irreducibility of a representation. Based on the MeatAxe one can give algorithmic solutions to more advanced tasks such as determining the endomorphism ring of a representation or the decomposition into a direct sum of indecomposable representations. Finally, we will discuss the condensation method, an important tool used in these applications. INI 1 15:00 to 16:00 Peter Cameron Finite permutation groups: the landscape post-CFSG In the 40 years since the Classification of Finite Simple Groups was first (tentatively) announced, our understanding of finite permutation groups has been radically transformed: previously untouchable classical problems have been solved, many new classifications produced, and applications to a wide area of mathematics, from number theory to computer science, have been developed. In this lecture I will outline the theory of finite permutation groups and explain how, using the O'Nan--Scott Theorem, CFSG can be brought to bear on various questions. In some cases, proofs avoiding CFSG have subsequently been found; but there are some interesting challenges where this has not so far proved possible. INI 1 16:00 to 16:30 Afternoon Tea 16:30 to 17:00 Ana Bernal On self-Mullineux and self-conjugate partitions The Mullineux involution is a relevant map that appears in the study of the modular representations of the symmetric group and the alternating group. The fixed points of this map are certain partitions of particular interest. It is known that the cardinality of the set of these self-Mullineux partitions is equal to the cardinality of a distinguished subset of self-conjugate partitions. In this talk we will see details on this, on the Mullineux map and I will show an explicit bijection between the two mentioned families of partitions in terms of the Mullineux symbol. INI 1 17:00 to 17:30 Cesare Giulio Ardito Classifying 2-blocks with an elementary abelian defect group Donovan's conjecture predicts that given a $p$-group $D$ there are only finitely many Morita equivalence classes of blocks of group algebras of finite groups with defect group $D$. While the conjecture is still open for a generic $p$-group $D$, it has been proven in 2014 by Eaton, Kessar, Külshammer and Sambale when D is an elementary abelian 2-group, and in 2018 by Eaton, Eisele and Livesey when D is any abelian 2-group. The proof, however, does not describe these equivalence classes explicitly. A classification up to Morita equivalence over a complete discrete valuation ring $\mathcal{O}$ has been achieved when $p=2$ for abelian $D$ with rank $3$ or less, and for $D=(C_2)^4$.In my PhD thesis I have done $(C_2)^5$, and I have partial results on $(C_2)^6$. I will introduce the topic, give some definitions and then describe the process of classifying these blocks, with a focus on the process and the tools needed to produce a complete classification. All the obtained data is available on https://wiki.manchester.ac.uk/blocks/. INI 1