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.

Three meetings are planned for 2017:

  • 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.

  • December 2017, RHUL (local organiser Brita Nucinkis)
    Cohomology of functor categories for topological groups

    We shall focus on new developments in the cohomology of totally disconnected topological groups as well as profinite and pro-p groups.

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) and Brita Nucinkis (Royal Holloway, University of London).