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
- Finite Groups
June 2022, Isaac Netwon Institute (hybrid) (organiser Nadia Mazza)
This meeting will build on the LMS ECR online lecture series (COVID Working Group Committee) on crowns, delivered by Gareth Tracey, and it takes place during the Isaac Newton Institute’s programme “Groups, representations and applications: new perspectives”. The theory of crowns in finite groups provide useful tools in the analysis of the group structure. In particular, they can be used to find the minimal number of generators of a finite group, and they also have applications related to the first cohomology group of a finite group. In this meeting, we propose to present some original techniques pertaining to the study of finite groups.
- Beauville groups
January 2022, online (organiser Anitha Thillaisundaram)
Beauville surfaces are a class of complex surfaces defined as products of Riemann surfaces with an action of a finite group. Finite groups with such an action are called Beauville groups. Beauville surfaces are interesting geometrical objects, providing, for instance, counterexamples to the Friedman-Morgan conjecture on diffeomorphic algebraic surfaces and enabling the constructions of exception collections of line bundles. Many of the properties of Beauville surfaces are determined by the properties of the corresponding Beauville groups. Hence studying Beauville groups translates to studying Beauville surfaces. Examples of Beauville groups are varied, from simple groups to more geometric groups. This meeting will introduce the connection between Beauville groups and Riemann surfaces, and then focus on recent developments involving Beauville groups, such as new links to groups acting on rooted trees.
- Locally analytic representations of p-adic groups
September 2021, Cambridge/hybrid/online (organiser Rachel Camina)
Locally analytic p-adic representations of p-adic groups are of great interest not just to representation theorists, but also to other areas: they arise naturally in number theory (where they form key objects in the mainly conjectural p-adic local Langlands correspondence) and in arithmetic geometry, with many constructions drawing inspiration from geometric representation theory.
This meeting, which can be viewed as a continuation of the introductory lectures of the LMS Autumn Algebra School 2020, aims to bring together young researchers and experts in the field to present current trends in research in this highly dynamic area. We hope to hold this meeting in a hybrid format but if this proves not to be possible, it will be online.
- Cohomology and geometry of infinite groups
April 2021, online (organiser Anitha Thillaisundaram)
Group cohomology is a classical subject with links to many areas of mathematics, such as representation theory and number theory. In this meeting, we will focus on the connections between cohomology of groups with geometric group theory. This meeting will build on the LMS ECR online lecture series (COVID Working Group Committee) on the proposed theme, delivered by Ilaria Castellano.
- Burnside rings for profinite groups
January 2021, online (organiser Nadia Mazza)
This meeting shall focus on the generalisation of Burnside rings from finite to profinite groups and their applications in representation theory in particular. In finite group theory, Burnside rings and the functoriality of mapping a group to its Burnside ring have led to very useful results, which have also been extended to fusion systems.
- Linear groups
September 2020, Lincoln (local organiser Anitha Thillaisundaram)
This meeting shall focus on linear groups and topics related to them. Linear groups play a central role in various subjects from representation theory, to Lie theory, to the theory of algebraic groups. Over finite fields they provide the bulk of the classification of finite simple groups through Chevalley’s theory.
- Cohomology and Mackey Functors for Profinite Groups
December 2019, Royal Holloway, University of London; to be held at Senate House, Central London (local organiser Brita Nucinkis)
This will be the final meeting in 2019 of the Research Group Functor Categories for Groups (FCG). Speakers at this meeting are N. Mazza (Lancaster), G. Corob Cook (Bilbao) and T. Weigel (Milano).
- Words in finite and profinite groups
September 2019, Lincoln (local organiser Anitha Thillaisundaram)
This meeting shall focus on a natural object in group theory: words. Several classical topics that are closely linked to the study of words are varieties of groups and group laws. Words also give rise to the concept of a verbal subgroup and is important in the study of group laws. The study of words has made significant contributions to our understanding of finite groups as well as profinite groups, as seen for example, by the influential theorem of Segal and Nikolov, that a finite index subgroup of a finitely generated profinite group is open.
- June 2019
Cambridge (local organiser Rachel Camina)
- Stable categories
April 2019, Lancaster (local organiser Nadia Mazza)
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.
- Graphs and groups
December 2018, Lancaster (local organiser Nadia Mazza)
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.
- Cohomology of functor categories for topological groups
September 2018, London (local organiser Brita Nucinkis)
We shall focus on the study of Hausdorff dimension for profinite groups, initiated by Abercrombie in the 90s.
- The category of totally disconnected locally compact groups
April 2018, Lincoln (local organiser Anitha Thillaisundaram)
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.
- (Pro-)fusion systems
September 2017, Lancaster (local organiser Nadia Mazza)
Introduced in the '70s, 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 the global theory of finite groups and the theory of profinite groups.
- Cohomology of functor categories for infinite discrete groups
May 2017, Galway, Ireland (local organiser Dieter Degrijse)
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 into the 2017 Groups in Galway conference.
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.
LMS Autumn Algebra School 2020
21st-25th September and 5th-9th October 2020
Jointly organised together with the research networks ARTIN and BLOC, the school was funded by the London Mathematical Society, the European Research Council, and the International Centre for Mathematical Sciences in Edinburgh. The school featured expository lecture series by early career researchers from among each research group.
Corresponding organisers of the Joint Research Group are: