Functor Categories for Groups

London Mathematical Society Joint Research Group

Groups are core to algebra, and their study now covers a wide range of techniques. Modern advances in group theory utilise categories to study properties for finite and infinite groups alike. Results obtained using categories such as fusion systems have allowed significant progress in the local-global study of finite groups, while Mackey functors and Bredon cohomology have been a major feature of the functorial study of groups, leading to major advances also in neighbouring areas such as algebraic topology, representation theory and K-theory in particular.

These functorial techniques, mainly developed for finite groups to date, have emerged in the study of infinite groups, and more recently in the study of profinite groups. This Research Group aims to bring together researchers representing the various subjects touched by functor categories for groups in order to incentivise future advances and stimulate new collaborations in the UK and Ireland.

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The next meeting will be held in April 2019:

  • April 2019, Lancaster (local organiser Nadia Mazza)
    Stable categories

    This meeting shall focus on stable (module) categories in group representation theory. There are different notions of "stable categories", but all have a common point: they are quotient categories, which are triangulated and have an extra multiplicative structure with a multiplicative identity. Stable categories are also intrinsically related to derived categories. The selected speakers are experts in varied aspects of stable and derived categories, and some of their recent results exhibit useful applications of stable categories.

Previous meetings of the Group were:

  • May 2017, Galway, Ireland (local organiser Dieter Degrijse)
    Cohomology of functor categories for infinite discrete groups

    The meeting shall focus on applications of functor cohomology and cohomology in categories to the study of infinite discrete groups focusing in particular on recent applications to homological stability and connections with equivariant stable homotopy theory, finiteness properties of groups, Mackey functors and Bredon cohomology. This meeting will be integrated in the 2017 ‘Groups in Galway’ conference.

  • September 2017, Lancaster (local organiser Nadia Mazza)
    (Pro-)fusion systems

    Introduced in the 70's, fusion systems are categories which model how non-conjugate subgroups in a Sylow p-subgroup of a given finite group can fuse, i.e. become conjugate, in the whole group. The study of fusion systems has led to significant advances and improvements of proofs in group theory, and also provided useful links with algebraic topology. The focus of the meeting will be on the use of fusion systems in the local to global theory of finite groups and the theory of profinite groups.

  • April 2018, Lincoln (local organiser Anitha Thillaisundaram)
    The category of totally disconnected locally compact groups

    This meeting, which is more of a masterclass in nature, shall focus on the category of totally disconnected locally compact groups and how they intersect with other areas, such as permutation groups, operator algebras and model theory. The totally disconnected compact groups are of course the profinite groups, whose influence is far-reaching. The locally-compact case has received much attention since the ground-breaking result of Willis in 1994.

  • September 2018, London (local organiser Brita Nucinkis)
    Cohomology of functor categories for topological groups

    We shall focus on the study of Hausdorff dimension for profinite groups, initiated by Abercrombie in the 90s.

  • December 2018, Lancaster (local organiser Nadia Mazza)
    Graphs and groups

    We shall focus on the interplay of graphs and groups, and how each of these structures can be used in the study of the other. Groups act on varied combinatorial structures and graphs in particular. On the other hand, given an abstract group, there are several ways to construct a graph associated to the group, such as π-products involution graphs, or commuting graphs. Recently, the approach of studying groups via associated graphs has been extended to hypergraphs, which leads to a useful generalisation of the theory.

The Research Group receives financial support from the London Mathematical Society and has therefore limited funds to reimburse travel expenses of UK-based students and young mathematicians. Please contact the organisers if you wish to apply for such reimbursements.

For UK-based mathematicians with caring duties the LMS has a Caring Supplementary Grant scheme which allows participants of meetings like ours to apply for help covering caring costs.

Organisers: Corresponding organisers of the Joint Research Group are Nadia Mazza (Lancaster University), Brita Nucinkis (Royal Holloway, University of London), Anitha Thillaisundaram (Lincoln) and Rachel Camina (Cambridge).