Timings: meeting at 13:00–16:30, followed by drinks and dinner.
Location: talks in Frank Adams Seminar Room, First Floor, Alan Turing Building (campus map).
13:00 Carlos Tapp Monfort (Rutgers University)
About Feit's conjecture
In 1979, Walter Feit proposed a conjecture asserting that for every finite group G and every irreducible character χ of G, there should be an element of G whose order is precisely the conductor of χ. This problem is still open. More recently, R. Boltje, A. Kleshchev, G. Navarro, and P. Tiep established a reduction theorem for the conjecture. In this talk, I will explain the associated inductive condition and present recent developments for finite groups of Lie type of small rank.
14:00 Marie Roth (University of East Anglia)
Unitriangularity of decomposition matrices: the simple adjoint finite groups of exceptional type
In 2020, Brunat–Dudas–Taylor showed that the decomposition matrix of the unipotent ℓ-blocks of a finite reductive group G in good characteristic has unitriangular shape. Their theorem holds under some conditions on the prime ℓ, in particular ℓ being good.
In this talk, we’ll discuss how to extend this result, firstly to ℓ bad (for any G simple) and then to other blocks, called isolated (for G simple of type G2 and F4). This work was part of my PhD thesis under the supervision of G. Malle and O. Dudas.
15:00 Tea & Coffee
15:30 Michael Bate (University of York)
Simple modules, invariants and structure of affine algebraic groups over a field.
Over algebraically closed fields, there are several reasons to study reductive algebraic groups. For example, they arise naturally when you study representation theory and geometric invariant theory, and they are nicely classified by root data. Over non-algebraically closed fields, the reasons for restricting attention to reductive groups are not quite so clear cut. In this talk I will try to convince you that there is some mileage in considering a wider class of algebraic groups, the so-called "pseudo-reductive groups". These were classified by Conrad-Gabber-Prasad in the 2010s, their simple modules were described by Bate-Stewart in the 2020s, and their subgroup structure and invariant theory is beginning to be investigated. I'll describe some aspects of the representation theory and use that to motivate a couple of problems I am thinking about at the moment.
Timings: meeting at 14:00–17:30, followed by dinner.
Location: talks in B3.03, Zeeman Building.
14:00 Pénélope Azuelos (University of Bristol)
Narrow Schreier graphs and virtual fibre subgroups
A finitely generated subgroup H of a finitely generated group G is a virtual fibre subgroup if G admits a finite index subgroup which surjects onto the integers and the kernel has finite index in H. This condition is very strong; it implies many nice properties of the subgroup and imposes a number of geometric properties on the Schreier graph of H∖G. In this talk, I will discuss the extent to which various geometric properties of the Schreier graph, including the number of ends, growth and 'narrowness' characterise virtual fibre subgroups.
15:00 Luca Sabatini (University of Warwick)
Expanders, diameter bounds and group properties
Let G be a finite group and let X be a symmetric generating set. Which properties of Cay(G,X) genuinely depend on the choice of X? It is known that having very large diameter, i.e. greater than or equal to |G|^\epsilon, is in fact a group property. Answering a question of Pyber and Szabo, we exhibit a group G with two generating sets X and Y of fixed cardinality such that Cay(G,X) is an expander graph (in particular with diameter O(\log|G|)), while Cay(G,Y) has super-polylogarithmic diameter. This is joint work with S. Eberhard.
16:00 Tea & Coffee
16:30 Emmanuel Breuillard (University of Oxford)
Spectral gaps in finite groups of Lie type
Cayley graphs of finite simple groups of Lie type, as opposed to say abelian or nilpotent groups, are expected to be expander graphs. I will discuss progress towards a conjecture that in bounded rank, finite simple groups are uniformly expanding. This means that the spectral gap depends only on the rank and not on the Cayley graph, nor the group. The method relies on transferring to finite characteristic a uniformity result for Zariski-dense subgroups of complex semisimple algebraic groups proved by means of some ingredients from diophantine geometry. This is joint work with Oren Becker.
Timings: meeting at 13:00–16:30, followed by drinks and dinner.
Location: talks in Lecture Theatre A (G23), Watson Building (R15 on campus map)
13:00 Patricia Medina Capilla (University of Warwick)
Crown-based powers and their applications
Initially discovered by Gaschutz in the 1950s, before later being extended by Dalla Volta and Lucchini in the 1990s, the theory of crowns in finite groups has a long and rich history, with numerous applications. Central to this theory is the observation that establishing generation results for a particular class of groups, known as crown-based powers, is often sufficient to derive corresponding results for all finite groups. In this talk, we will explore how this framework can be applied to a range of generation problems, highlighting in particular how the structure of a group’s chief factors determines its generation behaviour.
14:00 Coen del Valle (Open University)
The binary actions of sporadic groups
An action of a finite group is called binary if the permutation group it induces has relational complexity 2. We will begin by discussing the concepts of relational complexity and binary actions of permutation groups, before exploring some recent progress towards classifying the binary actions of the finite simple groups, with a particular emphasis on the recently completed classification of binary actions of the sporadic groups.
15:00 Tea & Coffee
15:30 Martin Liebeck (Imperial College London)
p-exceptional permutation groups and applications to character theory
Let p be a prime. A p-exceptional permutation group is a transitive group G for which p divides the order of a point stabilizer H, but divides none of the orbit sizes of H. I will describe various classification results for p-exceptional groups, and show how they apply to questions in character theory, such as generalisations of the Gluck-Wolf theorem.