Talks

8 May Bob Dabson (City)

Title: Periodic Modules and Perverse Equivalences

Abstract: Perverse equivalences, introduced by Chuang and Rouquier, are derived equivalences given by some combinatorial data. They are important in the representation theory of finite groups, particularly for groups of Lie type. They are of significance more generally in the representation theory of finite dimensional symmetric algebras. Grant has shown that periodic algebras admit perverse autoequivalences.  One can go further: I will demonstrate that certain periodic modules give rise to perverse equivalences, and vice versa, with a nice application to the setting of the symmetric groups.

Chair: Marina Anagnostopoulou-Merkouri

1 May Pavel Turek (Royal Holloway)

Title: Plethystic Murnaghan-Nakayama rule: proof using Loehr’s labelled abacus

Abstract: The Murnaghan-Nakayama rule describes how to multiply elements of two important bases of symmetric functions; the power sum symmetric functions and Schur functions. One of the many proofs of this result appeared in a paper by Loehr in 2010, where the author introduces a labelled abacus as a combinatorial model for antisymmetric functions $a_{\beta}$ and uses it to prove many standard rules for multiplying symmetric functions.

In the talk, we review this combinatorial model and some of the standard concepts in the theory of symmetric functions including a crucial binary operation called plethysm, which finds application in the representation theory of symmetric groups and general linear groups. Using Loehr’s labelled abacus we then prove the plethystic Murnaghan-Nakayama rule, a strengthening of the Murnaghan-Nakayama rule, where the power sum symmetric function is replaced by a plethysm of itself and a complete homogeneous symmetric function.

Chair: Marina Anagnostopoulou-Merkouri

24 April Jonathan Passant (Bristol)

Title: Which discrete structures can we count using groups?

Abstract: In the early 2010s Elekes and Sharir and later Guth and Katz gave a way to count the number of distances in a finite subset of the plane by counting partial symmetries in the Euclidean group.

I will give the motivation behind the so-called Elekes-Sharir-Guth-Katz framework and show how counting structures in groups gives allows us to count structures in the plane; present some nice results where this proof strategy leads to sharp bounds; then discuss problems where this approach fails, including recent progress on counting angles in the plane.

This talk will include work joint with Sam Mansfield and joint with Sergei Konyagin and Misha Rudnev.

Chair: Marina Anagnostopoulou-Merkouri

17 April Marco Fusari (Milan Bicocca)

Title: Cliques in derangement graphs

Abstract: Given a permutation group G, the derangement graph ΓG of G is the Cayley graph with connection set the derangements of G. In a recent paper of 2021 Meagher, Razafimahatratra and Spiga conjectured that there exists a function f : N → N such that, if G is transitive of degree n and ΓG has no k-clique, then n≤f(k). The conjecture has been proved for innately transitive groups, that are a generalization of primitive groups. Motivation for this work arises from investigations on Erdos-Ko-Rado type theorems for permutation groups.

(Joint work with P.Spiga and A.Previtali)

Chair: Hongyi Huang

20 March Eileen Pan (Warwick)

Title: Some primitive coset actions in G_2(q)

Abstract: The coset actions of almost simple groups have sparked much interest, and a lot has been done by various authors, but there remains much to explore. In this talk we consider G to be a finite group of type G_2, for example, G_2(q) for some prime power q and let H be a maximal-rank maximal subgroup in G. We describe the double cosets HgH for g in G and the corresponding intersections H \cap H^g. This in turn gives us the suborbit representatives and subdegrees of the action of G on the cosets of H. Along the way, we give a brief overview of some known results in algebraic and finite groups of exceptional Lie type and explain how they are used in approaching this problem.

Chair: Hongyi Huang

13 March Panagiotis Spanos (Ruhr-Universität Bochum)

Title: Nilpotent Groups, Transitive Graphs and Percolation

Abstract: In a Bernoulli percolation model on a graph G, each edge is independently labelled as "open" with probability p and "closed" with probability 1-p. The model is considered supercritical if the resulting percolated graph contains an infinite cluster with high probability. For Γ a finitely generated nilpotent group that does not grow linearly, we will construct a sequence of Γ-invariant Bernoulli percolation models, where the expected degree converges to 1. We naturally extend this construction to transitive graphs of polynomial growth. We will not discuss the probabilistic aspects in detail. Instead, our emphasis will be on the structure of nilpotent groups and their connections to transitive graphs. This presentation is based on joint work with Matthew Tointon.

Chair: Merlin Haith Rowlatt

6 March Asier Calbet Rípodas (Queen Mary)

Title: The asymptotic behaviour of sat(n, F)

Abstract: Given a family F of graphs, we say that a graph G is F-saturated if it is maximally F-free, meaning G does not contain a graph in F but adding any new edge to G creates a graph in F. We then define sat(n, F) to be the minimum number of edges in an F-saturated graph on n vertices. In 1986, Kászonyi and Tuza showed that sat(n, F)=O(n) for all families F and Tuza conjectured that for singleton families sat(n, F)/n converges. Tuza's Conjecture remains wide open. In this talk, I will discuss recent results about the asymptotic behaviour of sat(n, F), mostly in the sparse regime sat(n, F) ≤ n+o(n), in each of the cases when F is a singleton, when F is finite and when F is possibly infinite. Joint work with Andrea Freschi.

Chair: Hongyi Huang

21 February James Maxwell (Bristol)

Title: Connecting Tropical Geometry and Rigidity Theory

Abstract: Tropical varieties are polyhedral complexes with weights on the top dimensional pieces, satisfying a balancing (zero tension) condition. If the space of balanced weightings is one dimensional then we call the variety extremal. I will present results that describe how to translate the notion of extremality in the tropical setting to the concept of rigidity for the dual graph. This will then be related to understand decompositions of tropical varieties. This is joint work with Sean Dewar and Farhad Babaee.

Chair: Merlin Haith Rowlatt

7 February David Guo (Bristol)

Title: Uniform Dα property in polycyclic groups

Abstract: Given a polycyclic group, Wolf proved that polynomial growth is equivalent to being virtually nilpotent. A few years ago, Ana Khukhro, Alain Valette and Matthew Tointon defined a property of a residually finite group called ’uniform Dα’. Having uniform Dα is a weaker condition than having polynomial growth. It turns out that, for polycyclic groups, having uniform Dα is also equivalent to being virtually nilpotent. During this talk, we will go through some relevant definitions and the motivation behind this result.

Chair: Merlin Haith Rowlatt

24 January Stelios Stylianou (Bristol)