Siavash Lashkarighouchani
*University of St Andrews
***Homogeneous structures, their correspondence and construction
**

A homogeneous structure is a structure where any isomorphism between its finite substructures extends to an automorphism of the structure. This is a very strong symmetry condition that has great consequences. In this talk we cover the correspondence between classes of finite structures with the amalgamation property and homogeneous structures, and we will also look at the correspondence between the structural Ramsey theory of such classes and the topological dynamics of the automorphism group of the corresponding homogeneous structure. We will briefly look at the classification of the homogeneous permutations where the ideas of construction of homogeneous permutation structures is discussed. We shall end the talk with an overview of some interesting open questions.

Peter Cameron
*University of St Andrews
***Diagonal association schemes
**

A transitive permutation group is said to be AS-free if it preserves no non-trivial (symmetric) association scheme. This week, with Sean Eberhard, I have answered a question I asked fifteen years ago, showing that an AS-free group must be primitive and either 2-homogeneous or almost simple. The proof involves constructing an association scheme invariant under the one remaining class of primitive groups, the groups of simple diagonal type with at least four factors in the socle. AS-free groups which are not 2-homogeneous do exist, though they are not understood: examples include \(PSL(3,3)\) and \(PSL(3,3).2\) (degree 234), \(M_12\) (degree 1320), \(J_1\) (degrees 1463, 1540 and 1596), and \(J_2\) (degree 1800). Time permitting, I will explain the concepts involved.

Mohammed Aljohani
*University of St Andrews
***The synchronization property for diagonal groups with two factors
**

A permutation group \(G\) on finite set \(X\) is called synchronising if \(\langle G, f\rangle\) contains a constant function, where \(f\) is a non-permutation transformation. It is known that synchronising group is a primitive basic group. Therefore, according to the O’Nan-Scott theorem must be a subgroup in one of the groups in the following families; diagonal, affine and almost simple groups. In this talk, we will look at the synchronisation property for diagonal groups with two factors.

Raad Al-Kohli
*University of St Andrew
***coCF Groups
**

There's been a history of studying finitely generated groups by the type of formal language that the word problem belongs to, we will explore some of the results in this area. We will also discuss groups with language theoretic restrictions on the co-word problem. In particular, we will give attention to the co-word problem being context free.

Veronica Kelsey
*University of St Andrews
***The generating graph and maximal cocliques
**

Suppose a group \(G\) is \(2\)-generated. Then the generating graph of \(G\) is the graph whose vertices are the non-identity elements of \(G\), and with two elements joined by an edge whenever they generate \(G\). A set of vertices of a graph are coclique if no two are joined by an edge. We call a coclique maximal if it is contained in no larger coclique. In this talk, we will see examples and non-examples of maximal subgroups as maximal cocliques in the group's generating graph.

Luke Elliott
*University of St Andrews
***Polish semigroups
**

Polish spaces, and in particular Polish groups are heavily studied in descriptive set theory. There are many uniqueness results known for Polish topologies on various groups. We introduce the notion of a Polish semigroup and explore potential similar results for Polish semigroups.

Adan Mordcovich
*University of St Andrews
***Non-trivial maximal subgroups of almost simple finite groups.
**

In quite a few questions about the generation of finite groups, getting a handle on the maximal subgroups and their sizes can be very useful. This link can be seen fromthe fact that if a collection of elements of a group does not generate the whole group then they must lie in some maximal subgroups. Similarly understanding certain types of types of maximal subgroups of almost simple groups can be just as illuminating. These maximal subgroups, called non-trivial, do not contain the socle of the overgroup.

In this talk, we consider the problem of what the largest such maximal subgroups actually are.

Collin Bleak
*University of St Andrews
***A geometric interpretation of higher dimensional prime numbers.
**

Let \(n>1\) be an integer. We discuss the generation problem in the full monoid \(SL(n,N)\) of matrices with natural entries and determinant one. It is not hard to show that \(SL(2,N)\) is finitely generated, while Rivat was able to show in 2007 that \(SL(3,N)\) is not finitely generated. We give a complete geometric characterisation of irreducibles in the monoid \(SL(n,N)\). From this, it is easy to see that for all \(n>2\), \(SL(n,N)\) is not finitely generated. Joint with James Hyde.

Finn Smith
*University of St Andrews
***Minimal generating sets for the boolean matrix monoid
**

The boolean matrix monoid \(B_n\) is an important generalisation of certain well-understood monoids, including the full transformation monoid. However, it has a far more complicated and poorly understood structure. We would like to study this structure through computational methods, but in order to apply standard algorithms we require small generating sets for \(B_n\). I will describe how we found these for \(n\) up to \(8\).

Simon Jurina
*University of St Andrews
***The word and conjugacy problems in hyperbolic groups
**

In this talk we explore some of the geometric properties of decision problems in finitely presented groups and look at the state-of-the-art solutions to the word and conjugacy problems in finitely generated hyperbolic groups.

Saul Freedman
*University of St Andrews
***p-groups related to exceptional groups of Lie type
**

It is well-known that given any finite group, we can use standard representation theory to represent the group on some finite vector space, i.e., on some elementary abelian p-group. In 1978, Bryant and Kovács proved an analogue of this fact for p-groups that are not elementary abelian. Namely, if H is a subgroup of the general linear group GL(d,p), with d > 1, then there exists a p-group P such that Aut(P) induces H on the Frattini quotient of P. However, it is not known in general when we can choose P to be small, in terms of its exponent-p class, exponent, nilpotency class and order.

In this talk, we consider the representation theory of the finite simply connected versions of the (untwisted) exceptional groups of Lie type, in order to construct small related p-groups.

Fernando Flores Brito
*University of St Andrews
***A Guide Through The Congruence Lattice of End F_n (G)
**

Today I am going to teach you how to draw the congruence lattice of the endomorphism monoid of the free group-act. Congruences of this monoid can be classified in three types: ideal type, equality type, and complementary type. I shall explain what the previous means, revisit some old monoid friends, and see how they show up in our lattice. A rabbit may appear 🐇.

Gerard O'Reilly
*University of St Andrews
***Subalgebra Separability and Free Objects
**

I will introduce the finiteness condition of subalgebra separability using groups, where it is known as subgroup separability. I will then look at varieties of semigroups, asking whether the free objects within the variety are subalgebra separable.

Craig Miller
*University of St Andrews
***Right congruences of finite index
**

A semigroup is said to be f-Noetherian if every right congruence of finite index is finitely generated. In this talk, we will first discuss how the property of being f-Noetherian relates to other semigroup finiteness conditions; in particular, we will show that every finitely generated semigroup has this property. We will then exhibit some interesting examples of f-Noetherian semigroups that are not finitely generated. Finally, we will investigate whether the properties of being f-Noetherian and being finitely generated coincide for semigroups that lie in certain important classes, such as groups, semilattices, inverse semigroups and completely simple semigroups.

Hannah Sheats
*University of Wyoming, USA
***On Ryser's Conjecture and its variants
**

Let \(a\) be the independence number of a graph \(G\) with the edges colored in \(r\) colors. A component cover of \(G\) is a set of monochromatic components such that their union covers all vertices in \(G\) Ryser’s Conjecture states an upper bound for the size of the smallest component cover of \(G\) dependent on \(a\) and \(r\) In this talk we look at results concerning variations of Ryser's Conjecture obtained at an REU at Miami University in Ohio with Teresa Flaitz, Yigal Kamel, Cody Newman, and Louis DeBiasio.

Kamilla Rekvényi
*University of St Andrews
***Acyclic orientations and Turán graphs
**

Fix the number of vertices \(n\) and the number of edges \(m\) in a graph. Which graph maximizes the number \(a(n,m)\) of acyclic orientations of graphs with these given parameters? The "Hanging Curtains Conjecture", which asserts that if n and m are such that a Turán graph with n vertices and m edges exists, then this graph realises the maximum number of acyclic orientations; furthermore, the function \(f(m)=a(n,m)\) is concave for \(n\) fixed and \(m\) between the numbers of edges of successive Turán graphs.

In my talk I will present some recent progress towards the solution of this problem and share some further open problems connected to it.

Chris Russell
*University of St Andrews
***Counting finite congruence free semigroups
**

A congruence free semigroup has only two congruences - the universal congruence and the trivial congruence. A finite semigroup which is congruence free either be a simple group or a 0-simple semigroup with all subgroups trivial. The latter corresponds to binary matrices with all rows unique and all columns unique via the correspondence with Rees 0-matrix semigroups. I will demonstrate a method to count the isomorphism classes of these semigroups, which corresponds with counting these matrices up to permutations of the rows and columns.

Ashley Clayton
*University of St Andrews
***Fibered beasts and where to find them
**

A subdirect product of two semigroups S and T is a subsemigroup of the direct product for which the projections on to S and T are surjective. An important related construction of subdirect products is the fibered product of two semigroups, which arises as a pullback algebra given two epimorphisms onto a common semigroup. For algebras in congruence permutable varieties such as in the variety of groups, the two notions coincide. In this talk we explore to what extent subdirect products of free semigroups arise as fibered products, with particular emphasis on their finitary properties and the related decision problems. We also briefly introduce the notion of congruence length, and its role in discovering new fiber-like construction methods for subdirect products.

Miguel Del Río
*Campinas, Brazil
***Graph diagram groups
**

We study two families related to the Thompson groups F, T and V. The first of these is the Rearrangement Group of Fractals, introduced by Belk and Forrest in [1]. This is a family of groups that consist of homeomorphisms between self similar structures. We study Graph Diagram Groups, which provide a generalization of the Diagram Groups defined by Guba and Sapir in [2]. The idea of this new family of groups is to consider graph rewriting systems instead of string rewriting systems to define the diagrams. We will study the relation between these two families and take advantage of the diagram structure in Graph Diagram Groups to show certain properties on the elements in it.

Furthermore, we will study properties of Diagram Groups that remain true for Graph Diagram Groups. For example we study the conjugacy problem, product properties and the abelianization map on these groups.

[1] Belk, James and Forrest, Bradley: Rearrangement groups of fractals, arXiv preprint arXiv:1510.03133,2015.

[2] Guba, V. and Sapir, M., "Diagram Groups", Mem. Amer. Math. Soc., 130(620) viii+117pp., 1997.

Jana Bartoňová
*Brno, Czech Republic
***Polynomial closure of classes of regular languages
**

A polynomial closure is a certain operation on sets of regular languages. It plays a fundamental role in defining so-called concatenation hierarchies of regular languages. The natural question of decidability of the polynomial closure asking for an algorithm which decides whether a given regular language belongs to the polynomial closure of a given set of regular languages is hard to answer in general. I will shortly describe a method of finding a decidable characterization of the polynomial closure in some cases, following results by Branco, Pin (2009) and Place, Zeitoun (2017, 2018).

Justin Lanier
*Georgia Tech
***Normal generators for mapping class groups are abundant
**

Under what conditions do a group element and all of its conjugates form a generating set for the ambient group? Such an element is called a normal generator. We first survey some results about normal generators in a variety of groups. After introducing mapping class groups of surfaces, we provide a number of simple criteria that ensure that a mapping class group element is a normal generator. We then apply these criteria to show that almost every periodic element is a normal generator whenever genus is at least 3. We also show that every pseudo-Anosov mapping class with stretch factor less than √2 is a normal generator. Our pseudo-Anosov results answer a question of Darren Long from 1986. This is joint work with Dan Margalit.

Matt McDevitt
*University of St Andrews
***Consecutive patterns in permutations
**

We consider permutations ordered by the consecutive pattern involvement ordering, and describe a recent result on infinite anti-chains

Nayab Khalid
*University of St Andrews
***A presentation for some other rearrangement groups
**

In previous talks, I have defined rearrangement groups of fractals (Belk, Forrest 2016), constructed Thompson's group F as a rearrangement group, and subsequently developed an infinite presentation for it which stems directly from the geometric structure of the topological space. In this slightly ambitious talk, I will present the final bit of research of my PhD: a framework for similar infinite presentations for other rearrangement groups.

Peter Cameron
*University of St Andrews
***The Hall--Paige conjecture and an application
**

In 1955, Marshall Hall Jr. and Lowell Paige conjectured that the Cayley table of a finite group \(G\) has an orthogonal mate if and only if the Sylow 2-subgroups of \(G\) are trivial or non-cyclic. This was proved in 2009 by the combined efforts of Stuart Wilcox, Anthony Evans, and John Bray, but the final step (involving the sporadic group \(J_4\)) was never published.

Last year, an application emerged, a proof that primitive permutation groups of simple diagonal type with more than two factors in the socle are non-synchronizing. A paper has appeared on the arXiv including both a description of Bray's argument for \(J_4\) and a proof of the application (and a little bit more).

I will discuss these matters, starting from a minimal amount of background.

Sanming Zhou
*University of Melbourne
***The vertex-isoperimetric number of the incidence graphs of unitals and finite projective spaces
**

I will talk about some recent results on the vertex-isoperimetric number of the incidence graph of unitals and the point-hyperplane incidence graph of \(PG(n, q)\), where a unital is a \(2\)-\((n^3 + 1, n+1, 1)\) design for some integer \(n \ge 2\).

Joint work with Andrew Elvey-Price, Alice M. W. Hui and Muhammad Adib Surani.

Wolfram Bentz
*University of Hull
***Congruences on the product of transformation monoid
**

Congruences for transformation monoids, were first described in 1952, when Mal’cev determined the congruences of the monoid \(\mathcal{T}_n\) of all full transformations on a finite set \(X_n=\{1, \dots,n\}\). Since then, congruences have been characterized in various other monoids of (partial) transformations on \(X_n\), such as the symmetric inverse monoid \(\mathcal{I}_n\) of all injective partial transformations, or the monoid \(\mathcal{PT}_n\) of all partial transformations.

Although these results are about 60 years old, none of them have previously been generalized to products of two such monoids. Our work closes this gap by describing all congruences of \(\mathcal{T}_m \times \mathcal{T}_n\). As it turns out, the congruence structure of the factors is still visible in the congruences of the product, but the variations introduced by having an extra component adds a high level of technical complexity which accounts for the difficulty in achieving this result.

In addition to presenting the congruences of \(\mathcal{T}_m \times \mathcal{T}_n\), we will also address generalizations to products of \(\mathcal{PT}_n\), \(\mathcal{I}_n\), and matrix monoids, as well as generalizations to products with more than 2 factors.

This is a joint work with {\sc Jo\~{a}o Ara\'{u}jo} and {\sc Gracinda M.S. Gomes} (Lisbon).