Tuesday 21 January 2025, 14:00-15:00
Watson Building, B16
Abstract: TBA
Tuesday 4 February 2025, 14:00-15:00
Watson Building, B16
We present results that connect information theory and group theory via the so-called Ingleton inequality. The talk surveys some of the main known results.
Tuesday 18 February 2025, 14:00-15:00
Watson Building, B16
Tuesday 25 February 2025, 14:00-15:00
Watson Building, B16
We determine all reduced saturated fusion systems supported on a finite p-group of nilpotency class 2. As a consequence, we obtain a new proof of Gilman and Gorenstein's classification of finite simple groups with class 2 Sylow 2-subgroups.
Tuesday 4 March 2025, 14:00-15:00
Watson Building, B16
In this talk we will discuss the groups of even type as they appear in the project of Gorenstein, Lyons and Solomon.
Tuesday 11 March 2025, 14:00-15:00
Watson Building, B16
In their 1956 book Cartan and Eilenberg present results which tell us that the modular cohomology of a finite group G is equal to the set of stable elements in the modular cohomology of a Sylow p-subgroup of G. In this talk we will look at the groups SL2(ℤ/pn) for n > 1 and p an odd prime. Their cohomology is not yet known, however there is a way to obtain the cohomology, using a combination of tools from homological algebra, profinite group theory, and fusion systems. We will introduce the concepts used and show how they can facilitate the explicit computations.
Tuesday 18 March 2025, 14:00-15:00
Watson Building, B16
Decomposition Classes provide a natural way of partitioning a Lie algebra into finitely many pieces, collecting together adjoint orbits with similar Jordan decompositions. The current literature surrounding these tends to only cover certain cases - such as in characteristic zero, or under the Standard Hypotheses. Building on the prior work of Spaltenstein, Premet-Stewart and Ambrosio, we have managed to adapt many of the useful properties of decomposition classes to work in greater generality.
In this seminar talk, we will introduce decomposition classes for the Lie algebras of connected reductive algebraic groups - defined over arbitrary algebraically closed fields. We will then demonstrate some structure results about decomposition classes and their closures, which are referred to as decomposition varieties. This will allow us to highlight important connections to Lusztig-Spaltenstein Induction, and extend known results about the induction of nilpotent orbits to almost all applicable cases. Finally, we will touch on the importance of these concepts to the problem of classifying infinitesimally isolated nilpotent orbits.
Tuesday 25 March 2025, 14:00-15:00
Watson Building, B16
In the early 1960s, Higman and Sims proved that for any fixed prime p and large m, there are roughly p2/27 m3 nonisomorphic groups of order pm. The lower bound was obtained by counting the bilinear maps between two vector spaces. In 1978, Ol'shanskii showed the existence of a bilinear map such that the corresponding group of order pm has no abelian subgroup of order greater than p√(8m). In this seminar we see that picking a random bilinear map provides other wild p-groups, namely d-maximal groups and ab-maximal groups with large derived subgroups. This is joint work with S. Eberhard.
Tuesday 1 April 2025, 14:00-15:00
Watson Building, B16
Abstract: TBA
Tuesday 1 October 2024, 14:00-15:00
Watson Building, B16
I'll give a brief introduction to the current state of the art regarding asymptotic dimension of box spaces (and possibly dynamic asymptotic dimension of certain group actions). While the question of what the dimension of these objects can be when it is finite has been more or less settled, it is still open whether box spaces of finite-dimensional amenable groups must have finite asymptotic dimension. I will discuss a possible approach to this question involving a connection between asymptotic dimension and Banach densities. In particular, I will show a lemma (an application of Kerr and Szabó's ε-quasitiling theorem) which says, speaking somewhat loosely, that a box space of an amenable group has the same asymptotic dimension as the group except possibly on a subset with vanishing upper density (measure zero).
Tuesday 8 October 2024, 14:00-15:00
Watson Building, B16
The general linear group GLn(𝕜) over a field 𝕜 acts on the space 𝔤𝔩n(𝕜) of (n × n)-matrices by conjugation. The set 𝒩(𝔤𝔩n(𝕜)) of nilpotent matrices is preserved by this action and the group acts with finitely many orbits, with each orbit having a representative given by the Jordan normal form that is independent of 𝕜. In fact, more is true. The structure of the stabiliser of this nilpotent matrix is even independent of 𝕜.
The natural setting for these questions is the setting of connected reductive algebraic groups. One can ask to what extent these types of results hold for arbitrary split connected reductive groups. In this talk I will outline several reasonable ways to generalise the situation of GLn, both by changing GLn(𝕜) and the space 𝔤𝔩n(𝕜) on which it acts.
This is on-going joint work with Adam Thomas (Warwick).
Tuesday 15 October 2024, 14:00-15:00
Watson Building, B16
A group G is said to be free-by-cyclic if it contains a normal subgroup N which is free of finite rank and such that the quotient of G by N is infinite cyclic. Free-by-cyclic groups form a large class of groups which exhibits a range of interesting behaviour. A group G is said to be equationally Noetherian if any system of equations over G is equivalent to a finite subsystem. In joint work with Motiejus Valiunas, we show that all free-by-cyclic groups are equationally Noetherian. As an application, we deduce that the set of exponential growth rates of a free-by-cyclic group is well ordered.
Tuesday 22 October 2024, 14:00-15:00
Watson Building, B16
In general, it is hard to characterise "all subgroups" of a given group -- even hyperbolic groups still have many mysteries here. However, restricting the complexity in some way can make the problem tractable: subgroups of free groups, or of surface groups are not so bad, and cyclic subgroups don't cause too many problems. Two generators is a lot more than one, but progress can still be made: in 1979, Jaco and Shalen characterised the two-generator subgroups of fundamental groups of certain orientable three manifolds.
I will talk about recent work with Edgar Bering, Ilya Kapovich and Stefano Vidussi characterising the two-generator subgroups of mapping tori of free groups, using ideas from Feighn and Handel's proof of coherence for these groups.
Tuesday 5 November 2024, 14:00-15:00
Watson Building, B16
CW-complexes are topological spaces built inductively by gluing cells of different dimensions in specific ways. Given a group, one can canonically associate it with an aspherical CW complex, namely its classifying space. This gives us an equivalence between the category of groups and homomorphisms and the category of (aspherical) CW-complexes and cellular maps. We will try to better understand this dictionary through the lens of finiteness properties Fn, FPn and type F. I will then define the group cohomology and cohomological dimension. We will see how the cohomological dimension of a group G relates to the geometric dimension of G - the smallest possible dimension of a classifying space for G. A Z-structure on a group consists of a CW-complex X on which G acts freely and cocompactly and a 'nice' compactification G̅ of X. The topology of the "boundary" Z=X̅-X carries algebraic information about the group G. In particular, the topological dimension of Z equals the cohomological dimension of G. I will end by reviewing some known results and open questions about Z-structures.
Tuesday 12 November 2024, 14:00-15:00
Watson Building, B16
We call a group and a conjugacy class of involutions a k-transposition group if the class generates the group and any product of two elements in the class have order less than or equal to k. In 1970, Fischer investigated 3-transposition groups and as a result found three new sporadic simple groups. He also predicted the Baby Monster and Monster simple groups by looking at larger transposition groups. Since then, others have looked at certain transposition groups and produced classifications. In this talk, we will start by discussing the history before giving basic observations and results. The rest of the talk will be focused on the progress so far on simple, almost simple, and quasisimple groups and figuring out when they are or not 6-transposition groups. This is joint work with Chris Parker.
Tuesday 19 November 2024, 14:00-15:00
Watson Building, B16
Let n be a sufficiently large positive integer. A character is multiplicity-free if its irreducible constituents appear with multiplicity one. Wildon in 2009 and independently Godsil and Meagher in 2010 have found all multiplicity-free permutation characters of the symmetric group Sn. In this talk, we focus on a significantly more general problem when the permutation characters are replaced by induced characters of the form ρ↑Sn with ρ irreducible.
Despite the nature of the problem, I explain, combining results from group theory, representation theory and combinatorics, why this problem may be feasible and present a close to full answer. I also mention some of my (often surprising) results to questions about conjugate partitions, which naturally arise when solving the problem, and the remarkable complete classification of subgroups G of Sn, which have an irreducible character which stays multiplicity-free when induced to Sn.
Tuesday 26 November 2024, 14:00-15:00
Watson Building, B16
For a natural number n and p a prime, let Pn denote a Sylow p-subgroup of the symmetric group Sn. In 2017 E. Giannelli and G. Navarro proved that if χ is an irreducible character of Sn with degree divisible by p, then the restriction of χ to Pn has at least p different linear constituents. In this talk, we classify which characters of Sn whose restriction to Pn, has exactly p distinct linear constituents (not counting multiplicity). Most of the talk will focus on the p = 2 case. For odd primes, the classification follows more directly from results by S. Law and E. Giannelli.
This is joint project with Stacey Law.
Tuesday 10 December 2024, 14:00-15:00
Watson Building, B16
In this talk, we give a brief overview of s-arc-transitive graphs and show how their study in the case of cubic (3-regular) graphs reduces to a generation problem for finite almost simple groups. We then discuss current progress towards solving this generation problem for PSL(n,q) when n is sufficiently large and q is prime to 6.