Talks

6 November Fred Tyrrell (Bristol)

Title: New Lower Bounds for Cap Sets

Abstract: A cap set is a subset of F_3^n with no solutions to x + y + z = 0 other than when x = y = z, or equivalently no non-trivial 3-term arithmetic progressions. The cap set problem asks how large a cap set can be, and is an important problem in additive combinatorics and combinatorial number theory. In this talk, I will introduce the problem, give some background and motivation, and describe how I was able to provide the first progress in 20 years on the lower bound for the size of a maximal cap set. Building on a construction of Edel, we use improved computational methods and new theoretical ideas to show that, for large enough n, there is always a cap set in F_3^n of size at least 2.218^n. I will then also discuss recent developments, including an extension of this result by Google DeepMind.

30 October Karel Devriendt (Oxford)

Title:  Spanning Trees, Effective Resistances and Curvature on Graphs

Abstract: Kirchhoff's celebrated matrix tree theorem expresses the number of spanning trees of a graph as the maximal minor of the Laplacian matrix of the graph. In modern language, this determinantal counting formula reflects the fact that spanning trees form a regular matroid. In this talk, I will discuss some consequences of this perspective for the study of a related quantity from electrical circuit theory: the effective resistance. I will give a new characterization of effective resistances in terms of a certain polytope and discuss applications to recent work on discrete notions of curvature based on the effective resistance.

Chair: Marina Anagnostopoulou-Merkouri

23 October Debmalya Bandyopadhyay (Birmingham)

Title: Monochromatic tight cycle partitions in edge-coloured complete k-graphs 

Abstract: Let K_n^{(k)} be the complete k-uniform hypergraph on n vertices. A tight cycle is a k-uniform graph with its vertices cyclically ordered so that every k consecutive vertices form an edge, and any two consecutive edges share exactly k-1 vertices. A result by Bustamante, Corsten, Frankl, Pokrovskiy and Skokan shows that all r-edge coloured K_{n}^{(k)} can be partition into c_{r,k} vertex disjoint monochromatic tight cycles. However, the constant c_{r,k} is of tower-type. In this work, we show that c_{r,k} is a polynomial in r. 

Chair: Merlin Haith Rowlatt

16 October Bim Gustavsson (Birmingham)

Title: Counting linear constituents of the restriction of almost hook characters

Abstract: For a natural number n, let P_n denote a Sylow p-subgroup of the symmetric group S_n. In 2017 E. Giannelli and G. Navarro proved that if \chi is an irreducible character of S_n with degree divisible by p, then the restriction of \chi to P_n has at least p different linear constituents. For p=2, the only characters of S_n of odd degree are hook characters, hence the restriction to P_n of any non-hook character have at least 2 linear constituents. The so-called almost hook characters are important when we want to determine the characters of S_n whose restriction to P_n have exactly two linear constituents. This talk will focus on linear constituents of the restriction of almost hook characters of S_n to P_n, when p=2 and n is some power of 2. If time allows, we will also compute the multiplicity of these linear constituents.

Chair: Merlin Haith Rowlatt

9 October Sean Dewar (Bristol)

Title: Edge-length preserving embeddings of graphs between normed spaces.

Abstract: A finite simple graph G=(V,E) is said to be (X,Y)-flattenable if any set of induced edge lengths from an embedding of G into a normed space Y can also be realised by an embedding of G into a normed space X. This property, being minor-closed, can be characterized by a finite list of forbidden minors.

In my talk, I will establish fundamental results about (X,Y)-flattenability. Firstly, I will discuss how the Banach spaces L2 and L-infinity serve as two natural extreme spaces of flattenability. With this, I'll provide a complete characterization of (X,Y)-flattenable graphs for the specific case when X is 2-dimensional and Y is infinite-dimensional. I will also describe sufficient conditions under which (X,Y)-flattenability implies independence with respect to the associated rigidity matroids for X and Y.

Chair: Merlin Haith Rowlatt

2 October Ryan Lam (Bristol)

Title: The Injective Generation Condition

Abstract: Let A be a finite dimensional algebra over an algebraically complete field. Rickard proved that if A satisfy a condition which we call "injective generation", then the finitistic dimension conjecture holds for A. While the conjecture and its consequences are interesting on their own, the talk will focus on introducing and verifying (if time allows) the injective generation condition of a particular algebra A, with methods introduced by Goodearl, Huisgen-Zimmermann and many others.

Chair: Merlin Haith Rowlatt

25 September Alvaro Gutierrez Caceres (Bristol)

Title: Determinant of the distance matrix or a tree.

Abstract: Let T=([n],E) be a tree and M be the matrix whose (i,j) entry is the distance between i and j in T. The determinant of this matrix has an easy expression ±(n-1)2^(n-2) that does not depend on the tree structure. It is our goal in this talk to develop the right combinatorial tools that answer the question "what is this determinant counting?" 

Chair: David Guo

18 September Hongyi Huang (Bristol)

22 May Jamie Mason (Birmingham)

Title: (2,p)-Generation of the Suzuki Groups and other Exceptional Groups of Lie Type

Abstract: Since Steinberg's original result in the 60s that all known simple groups are generated by two elements, much work has been done in the field of group generation. I will give a brief outline of some of the major results and other background in the field and then go on to discuss the specific problem of generating exceptional groups of Lie type by an involution and an element of order p. Hopefully, if time permits, I will give an outline of how one actually proves (2,p)-generation for a family of groups, namely the Suzuki groups. 

Chair: Marina Anagnostopoulou-Merkouri

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)