# Seminar Archive

Semester 1 programme, Session 2023-24

**Thursday 14th September 2023**: Jack Allsop (Monash University)

**Row-Hamiltonian Latin squares and perfect 1-factorisations**

Abstract: A Latin square of order n is an n x n matrix of n symbols, such that each symbol occurs exactly once in each row and column. Let L be a Latin square of order n. Each pair of distinct rows of L forms a 2-line permutation. If this permutation is a single n-cycle, for any choice of rows, then L is called row-Hamiltonian. Each Latin square has six conjugate Latin squares, obtained by uniformly permuting the coordinates of its (row, column, symbol) triples. Let \nu(L) denote the number of conjugates of L which are row-Hamiltonian. It is known that \nu(L) \in \{0, 2, 4, 6\} and for each m \in \{0, 2, 6\} there are known infinite families of Latin squares with \nu = m. We construct the first known infinite family of Latin squares with \nu=4.

A 1-factorisation of a graph is a partition of its edge set into 1-factors. A 1-factorisation is perfect if the union of edges in any pair of its 1-factors forms a Hamiltonian cycle. A perfect 1-factorisation of the complete bipartite graph K_{n, n} is equivalent to a row-Hamiltonian Latin square of order n. Our family of Latin squares with \nu=4 allows us to build the eighth known infinite family of perfect 1-factorisations of complete bipartite graphs.

**Thursday 28th September 2023**: Rosemary Bailey (St Andrews)

**Designs on strongly regular graphs**

Abstract: I will talk about experiments where each treatment is applied to some of the vertices of a strongly regular graph. The corresponding Bose-Mesner algebra has three common eigenspaces. I assume that these are also the eigenspaces of the variance-covariance matrix of the responses on the vertices.

There are two (conflicting!) desirable properties of the design that do not depend on the eigenvalues of this matrix. I will describe both of these in the context of two familiar strongly regular graphs.

**Thursday 12th October 2023**: Scott Harper (St Andrews)

**Bases and generating sets for finite groups**

**Thursday 26th October 2023**: Bromlyn Cameron (St Andrews)

**Transition chains and Thompson’s groups**

Abstract: In the 1960s R. Thompson defined a family of three finitely presented groups of homeomorphisms of the Cantor set. Two of these groups, T and V, were the first known examples of finitely presented infinite simple groups. We are interested in the other group – Thompson’s group F – and more specifically groups of PL_+(I) homeomorphisms which do not contain an isomorphic copy of F. In this talk we will introduce transition chains and discuss their place in the study of F-less groups of homeomorphisms.

**Thursday 9th November 2023**: Florent Hivert (University Paris-Saclay / Orsay)

**Diagrammatic for Okada monoid and algebra**

Abstract: It is well known that the Young lattice is the Bratelli diagram of the symmetric groups expressing how irreducible representations restrict from S_N to S_{N-1}. In 1975, Stanley discovered a similar lattice called the Young-Fibonacci lattice which was realized as the Bratelli diagram of a family of algebras by Okada in 1994. In this talk we will present a combinatorial model for the Okada algebra and the associated monoid using a labeled version of the arc diagrams of the Jones monoid and the Temperley-Lieb algebra. We prove that the cardinality of the Okada monoid and dimension of the Okada algebra is n! in full generality (it was only proven in the semisimple case by Okada). In particular, we interpret the natural bijection between permutations and labeled arc diagrams as an instance of the Robinson-Schensted-Fomin correspondence associated to the Young-Fibonacci lattice. This has a lots of algebraic consequences: the aperiodicity of the Okada monoid and its Green relations as well as the cellularity of the Okada algebra. (Joint work with Jeanne Scott.)

**Thursday 23rd November 2023**: Reinis Cirpons (St Andrews)

**Computing in Free Bands**

Semester 2 programme:

**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.

**Week 9 – Thursday 21st March 2024**: Pilar 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**

**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**

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**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.

**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.