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

3-7 juin 2019

Institut Mittag-Leffler - Djursholm (Suède)

### http://ewmems2019mli.sciencesconf.org

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**

Lieu de la conférence