# Seminars

In the academic year 2023-24, the Algebra and Combinatorics Seminar will run on Thursdays of odd-numbered weeks, at 1-2pm.  The venue will be Maths Lecture Theatre D.

Our Semester 2 programme is as follows.

Week 1 – Thursday 18th January 2024: Peter Cameron (St Andrews)

Covers of sets of groups

Abstract:  Let S be a set of finite groups. A cover of S is a finite group G containing a copy of each group in S; it is minimal if no proper subgroup of G is an S-cover, and minimum if no S-cover has smaller order than G.  The topic was suggested to me by Hamid Reza Dorbidi from Jiroft. Subsequently David Craven (Birmingham) and Benjamin Sambale (Heidelberg) have contributed. But many open problems remain.  I will talk about a few of the results and problems, including the following.

•  We can describe completely the smallest abelian group containing a given set of abelian groups; but we do not know whether it is the minimum cover.
• The set of groups of order 4 has only finitely many minimal covers, but the set of groups of order 8 has infinitely many.
• A minimum cover of a set of two non-abelian simple groups is either their direct product or a simple group.

Week 3 – Thursday 1st February 2024: Louis Theran (St Andrews)

Title: Orthogonal representations of graphs revisited

Abstract:  An orthogonal representation (OR) of a graph G with n vertices is a configuration of n vectors in a d-dimensional inner product space so that the vectors associated with vertices i and j are orthogonal whenever {i, j} is not an edge of G. ORs, and their general position counterparts, GORs were introduced by Lovász to study the Shannon capacity of a graph, but they have many other applications. I’ll discuss some of these (old and new), and, time permitting, an algebra-geometric approach to constructing a GOR.

Week 5 – Thursday 15th February 2024: Jung Won Cho/Pierre Zhou

Title (Jung Won Cho): Non-finitely presented semigroups

Abstract:  In 1975, B.M. Schein proved that free inverse semigroups are not finitely presentable as semigroups. Motivated by this result, we will characterise finite presentability of subsemigroups of the monogenic free inverse semigroup. We will also see how Schein’s transformations can be used to answer James East and Carl-Fredrik Nyberg-Brodda’s question on whether a free regular *-semigroup is finitely presentable.

Title (Pierre Zhou): Determine the atomicity of a family of finitely based permutation classes

Abstract:  Permutations, when interpreted as models of some first-order theory whose signature consists of two linear orders, are very often viewed as words, hence can be partially ordered by some arguably ‘natural’ subword involvement (or, containment) order. The downward closed sets of permutations under this particular order are called permutation classes. A permutation class is said to be atomic if it cannot be written as the union of two proper subclasses. Model-theoretically, this is equivalent to having the so-called joint embedding property (JEP). In this talk, we will determine the atomicity of a family of finitely based permutation classes by first inspecting a concrete introductory example. Then, after a quick detour into the notion of symmetries, we shall be able to prove that every member of this family is atomic by showing that the image of each class under the described symmetries has the joint embedding property. This result is part of a work-in-progress classification theorem.

Week 7 – Thursday 7th March 2024: Tom Coleman (St Andrews)

Title: Group-embeddable monoids and graphs

Abstract:  Frucht’s theorem states that every finite group arises as the automorphism group of some simple undirected graph. Generalizations have been made in a number of different directions; both in the nature of the algebraic object on one side and the combinatorial object on the other. For instance, any monoid arises as the endomorphism monoid of some graph, and any finite group arises as the automorphism group of a strongly regular graph. In particular, de Groot and Sabidussi independently proved that any group (not necessarily finite) arises as the automorphism group of a graph.

If a monoid is group-embeddable, then it can be embedded in some symmetric group and viewed as a monoid of permutations. A permutation on the vertices of a graph G may preserve edges but not non-edges; this is known as a bimorphism of G. In this talk, we will demonstrate that every group-embeddable monoid arises as the bimorphism monoid of some graph, generalising the result of de Groot/Sabidussi. This is the result of joint work with Isaac Dilley.

Week 9 – Thursday 21st March 2024Pilar Duque Paez/Joseph Edwards (St Andrews)

Title (Pilar Duque Paez): The embedding of certain linear and abelian groups in finitely presented simple groups

Abstract:  In 1984 Elizabeth A. Scott defined a method for constructing finitely presented infinite simple groups. As a consequence of her construction, it was shown that GL(n, Z), the group of nxn matrices with integer entries and determinant 1 or -1, embeds in the group of automorphisms of the m-ary tree, Aut(T_m). This is a much-cited result of Brunner and Sidki from 1998. In this talk, we will give a brief description of Scott’s construction, resulting in the embedding of certain linear and abelian groups in finitely presented simple groups.

Title (Joseph Edwards): Solving the Word Problem: Modern Data Structures for the Knuth-Bendix Procedure

Abstract: In 1970, Knuth and Bendix devised an algorithm for “term rewriting systems” that can decide when two terms composed of variables and operators are equal, given a set of rewriting rules. Fifteen years later,  Kapur and Narendran applied this algorithm to string rewriting systems, proving several results regarding the existence, uniqueness, and usefulness of such systems. In this talk, we discuss how string rewriting systems can sometimes be used to solve the word problem for semigroups, and investigate how the classical Knuth-Bendix algorithm can be improved by taking advantage of modern data structures.

EXTRA SEMINAR!  Note non-standard day and venue (Maths Lecture Theatre C)

Week 10 – Tuesday 26th March 2024, 1-2pm: David Beers (Oxford)

Title: Level Sets of Persistent Homology for Point Clouds
Abstract: Persistent homology is an operation which takes as input a point cloud and returns a collection of intervals, in an attempt to capture the topology of the space the point cloud was sampled from. How descriptive is this operation? We address this question by studying the dimension of the level sets of the persistence map. In particular, we discuss upper and lower bounds for this dimension and describe connections to rigidity theory. No background in homology theory will be assumed.

Week 11 – Thursday 4th April 2024: Dorte Behrens/ Murray Whyte (St Andrews)

Title (Dorte Behrens): Homogeneous oriented two-graphs

Abstract: In 1954 Fraïssé wrote his theorem on amalgamation classes and the existence of a homogeneous structure that forms their limit. Since then classifications of the homogeneous graphs (Lachlan & Woodrow, 1980), digraps (Cherlin, 1998) 3-hypergraphs (Lachlan & Akhtar, 1995) and others have followed. In this talk we will be considering the homogeneous oriented two-graphs.

Title (Murray Whyte): Short presentations for transformation monoids

Abstract:  Many widely-loved transformation monoids, such as the full transformation, symmetric inverse and partial transformation monoids, contain a copy of the symmetric group of the same (here, finite) degree; any presentation for one of these monoids contains a presentation for such a symmetric group, plus a collection of additional defining relations. We’re interested in finding small presentations for these monoids ‘modulo the symmetric group’. In other words, we ask what the minimum number of these additional defining relations is, over all possible presentations of our considered monoids.

I’ll exhibit what I understand to be the smallest-known presentations for these monoids with respect to the number of additional relations — in other words, finding upper bounds in response to this question. I’ll also discuss the pursuit of lower bounds: considering what any collection of these additional relations must be able to ‘do’, so as to conclude there must be a certain number of them in any presentation.

EXTRA SEMINAR!  Note non-standard day and time (but standard venue: Theatre D)

Week 12 –  Monday 8th April 2024, 4-5pm: Charles Cox (Bristol)

Title: Wreath products and invariable generation

Abstract: We’ll begin with a gentle introduction to
(1) a wreath product, and
(2) the concept of a group having an invariable generating set.
By mostly working with well-known or concrete examples (based in some way on the integers). There are then two proofs, whose basic ideas I’ll aim to capture for an audience new to both (1) and (2). I hope to illustrate where the proofs have come from and why they are natural.