Seminars
Seminars take place in EBS 2.65 from 11am. Tea and cakes will be served in the same room from 10.30am.
Monday 6th July
Speaker: Peiran Wu (KU Leuven)
Title: Formalising group theory: a case study in irredundant bases
Abstract: The computer-assisted formalisation of mathematics is a process that sees proofs being written as computer code so that they can be rigorously checked by software. A landmark example is the formalisation in 2013 of the proof of the odd order theorem in Coq (now Rocq). Given the considerable community momentum and the comprehensive library of mathematics written in Lean, a relatively new theorem prover, it is arguably easier now than ever to formalise both classical and modern results in group theory, though many obstacles remain.
In this talk, I will demonstrate how I formalised a recent result concerning the maximum irredundant base size. An irredundant base for a permutation group is a sequence of points that yields a strictly descending chain of pointwise stabilisers, terminating at the trivial subgroup; the concept arises from the Schreier–Sims algorithm introduced in 1970. I will present – in natural language and in Lean – a formula for the maximum irredundant base size of a wreath product G wr H in product action, where every point stabiliser in G has exactly one fixed point.
Along the way, I will reflect on the motivation for formalisation and its practical challenges. I will also sample recent developments in the formalisation community and consider where the formalisation of group theory might be (or should be) headed.
Tuesday 7th July
Speaker: Saul Freedman (Colorado State University)
Title: Relational complexity: permutation groups and relational structures
Abstract: The relational complexity of a permutation group G on a set Ω is a statistic measuring the minimal “complexity” of certain combinatorial/model-theoretic objects (homogeneous relational structures defined on Ω) with automorphism group G. Equivalently, the relational complexity of G is a measure of the way in which the orbits of G on k-tuples of points of Ω, for various k, determine the action of G on arbitrarily long tuples.
Following a gentle introduction to these concepts, we will present new results, joint with Veronica Kelsey and Colva Roney-Dougal, on the relational complexity of almost simple linear groups, and on a related statistic: the minimal number of relations of a homogeneous relational structure with automorphism group G.
Wednesday 8th July
Speaker: Alexei Vernitski (University of Essex)
Title: Automated reasoning, how to train finite automata, and the Andrews-Curtis conjecture
Abstract: The Andrews-Curtis conjecture is an open problem on the borderline of algebra and topology. Due to its computational nature, it also attracts researchers who study machine learning and automated reasoning. In this talk I will introduce automated reasoning on simple examples. I will describe a new machine learning technique, based on finite automata, which we developed with group-theoretical applications in mind. I will explain how they combine and compare with a special-purpose search algorithm which we designed for working with the Andrews-Curtis conjecture.
Thursday 9th July
Speaker: Gerald Williams (University of Essex)
Title: Incidence graphs of generalized polygons and star graphs of group presentations with cyclic symmetry
Abstract: A generalized polygon is a point-line incidence structure that includes projective planes (generalized 3-gons). Incidence graphs of generalized m-gons are connected bipartite graphs of diameter m and girth 2m. Associated to any group presentation is a graph called the star graph, which encodes structural information about the group defined by the presentation. Transitional behaviour can occur for groups defined by presentations whose star graph components are incidence graphs of generalized polygons; such presentations are called “special”. A cyclic presentation of a group is a type of group presentation that admits a cyclic symmetry. In this talk I will discuss joint work with Ihechukwu Chinyere in which we classify the special cyclic presentations.
Friday 10th July
Speaker: Shuo Feng (University of Essex)
Title: The decomposition of ∆(a, b) ⊗ ∇(b, a)
Abstract: In this talk, I will discuss the decomposition of the tensor product of the Weyl module Δ(a,b) tensor product it’s dual ∇(b,a) for sl_3. The main idea is to use the geometry of alcoves to track the Weyl factors appearing in the tensor product. By introducing a decomposition tree and the notion of Weyl factor distance, we can organize the multiplicities and determine how the Weyl factors combine into tilting modules. I will focus mainly on the cases where the highest weight lies in Alcove 2 and Alcove 5 in this talk.
Social activities
Tea and cakes will be served from 3pm every day in 4SW.6.18.
Note the change of day: On Tuesday afternoon, we will have a party going climbing at the nearby bouldering wall. If you want to join in, please e-mail Alastair Litterick.
Participants
- Michael Bate
- Jessica Claridge
- Dibyendu Das
- Elena Denisova
- Heiko Dietrich
- Shuo Feng
- Saul Freedman
- Lewis Groves
- Sema Gunturkun
- Ameen Hasan
- Hongyi Huan
- Kanwal Khalid
- Mikko Korhonen
- Melissa Lee
- Lufeng Li
- Alastair Litterick
- Runqi Liu
- Joseph Malbon
- Jesus Martinez-Garcia
- Bernhard Muehlherr
- Eamonn O’Brien
- Rachel Pengelly
- David Penman
- Cheryl Praeger
- Aluna Rizzoli
- Colva Roney-Dougal
- Charlotte Satchwell
- Damian Sercombe
- Cristiano Spotti
- David Stewart
- Joel Summerfield
- Harvey Sykes
- Jay Taylor
- Adam Thomas
- Alexei Vernitski
- Matthew Westaway
- Gerald Williams
- Billy Woods
- Peiran Wu