Modern theory of group actions and the special role of finite simple groups

3-7 juin 2019
Institut Mittag-Leffler - Djursholm (Suède)

The topic of the summer school is motivated by recent applications of group actions to various questions in algebra, geometry, number theory and computer science which have given rise to the development of new theoretical results as well as algorithms for computer algebra software. The theme is approachable for young researchers via the theory of permutation groups and there are many open questions, both on the theoretical side and with regards to applications. Just to illustrate the variety of problems that are related to this: The theory of quasiprimitive permutation groups (a larger class than the primitive groups) developed because several analyses of graph symmetry exploited reductions which often led to quasiprimitive group actions, but almost never to primitive group actions. The theory of maximal factorisations of simple groups developed as part of a strategy to classify all maximal subgroups of finite symmetric and alternating groups. The resulting classification for the almost simple groups provides one of the most useful resources for applications of the finite simple group classification which involve group actions. The theoretical quality of algorithms for permutation groups or matrix groups is measured by complexity. One of the main tools for these complexity analyses is statistical group theory on finite classical groups and alternating groups (mainly proportions of particular kinds of elements). Sharp estimates for such proportions run hand in hand with accurate estimates for the failure probabilities of Las Vegas randomised algorithms, and they lead to faster algorithms. We will introduce several aspects of modern permutation group theory in three mini courses and supplement these with three invited spotlight lectures on related ongoing research. The topic is broad-based because of the importance of studying symmetrical structures. Objects exhibiting symmetry are relevant in many areas of science, and there are new powerful tools available exploiting the finite simple group classification, through the theory of group actions and of randomised algorithms. Therefore, it is suitable for a summer school. Young researchers working in diverse fields will benefit from deeper knowledge about the current research activities in these areas. The mixture of purely theoretical topics, applications and a variety of computational aspects will give a broad overview and allows many potential participants to extend and complement their existing knowledge.
Discipline scientifique : Mathématiques - Théorie des groupes

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