Fernando Flores Brito and Michael Torpey
*University of St Andrews
***
**

**Fernando Flores Brito**

**Title:**
Congruences of the endomorphism monoid of a free group-act

**Abstract:**

**Michael Torpey**

**Title:**
Congruences of the martition monoid

**Abstract:**
The partition monoid of degree n is the set of all equivalence relations on 2n
points, under an interesting concatenation operation. In this talk I will
define the partition monoid and some important submonoids, describe its
congruences, and explain some of the computational ways in which we discovered
them.

Ashley Clayton, Craig Miller, Chris Russell
*University of St Andrews
***
**

**Ashley Clayton**

**Title:**
Counting isomorphism classes of subdirect products of infinite semigroups

**Abstract:**
A subdirect product C of two algebraic structures A and B is a subalgebra of
the direct product AxB such that the projection maps from C to A and from C to
B are surjective. We can ask for what algebraic properties P does the statement
"C has property P if and only if A and B have property P" hold, and extend this
to subdirect/direct products of countably many algebraic structures. We
consider such statements, and in particular show that for A,B infinite
semigroups containing an isomorphic copy of the free monogenic semigroup, that
the property "has countably many subsemigroups" is not preserved under taking
direct products.

**Craig Miller**

**Title:** Presentations of \(M\)-acts

**Abstract:**
This talk will first introduce the theory of \(M\)-acts and then we will discuss
presentations of acts. Our main result provides a characterisation of those
monoids for which the class of finitely generated \(M\)-acts coincides with the
class of finitely presented \(M\)-acts.

**Chris Russell**

**Title:**
E-unitary inverse semigroups in GAP

**Abstract:**
The class of E-unitary inverse semigroups is an important one in inverse
semigroup theory due to McAlister’s covering theorem which states that every
inverse semigroup is an idempotent-separating homomorphic image of an E-unitary
inverse semigroup. The structure of these semigroups is more easily understood
when they are represented as McAlister triple semigroups. I have spent some
time implementing these objects into the Semigroups package of GAP and I aim to
provide an overview of what they are, the work I have done and my future plans.

Horacio Guerra and Simon Jurina
*University of St Andrews
***The power graph of a torsion-free group
**

The power graph of a group has vertices \(x\) and \(y\) adjacent if one is a power of the other; the directed power graph has an arc from \(x\) to \(y\) if \(y\) is a power of \(x\). It is known that, for finite groups, the power graph determines the directed power graph up to isomorphism; this fails for infinite groups. We show that, for some classes of torsion-free groups such as nilpotent groups of class 2, the assertion holds; while, for special groups such as direct sums of \(\mathbb{Z}\) or \(\mathbb{Q}\), even stronger results hold.

Nayab Khalid, Adan Mordcovich, Wilf Wilson
*University of St Andrews
***
**

**Nayab Khalid**

**Title:**
Topological Properties of Connected Rearrangement Groups

**Abstract:**
I will define connected rearrangement groups and study some of their dynamical
and topological properties. In particular, when does the generating set for the
group correspond to the basic open sets of the self-similar topological space?

** Adan Mordcovich**

**Title:**
Probabilistic Generation of Simple Finite Groups

**Abstract:**
Given a finite group we can calculate the probability that two elements picked
at random generate the whole group. Restricting our focus to finite simple
groups, we discuss bounds on the probability and various results.

**Wilf Wilson**

**Title:**
Maximal subsemigroups via independent sets

**Abstract:**
I will explain how maximal subsemigroups of a monoid can be described and
counted in terms of an associated graph. I will do this through examples, such
as the monoid of all order-preserving transformations, and the Jones monoid.

Laura Ciobanu Radomirovic
*Heriot-Watt University
***Plants, languages and groups
**

In the 1960s Lindenmeyer introduced a class of grammars and languages, called L systems, whose goal was to model the growth of plants and other organisms. It turns out that these languages also describe lots of important sets that naturally occur in group theory. The set of primitive words in the free group of rank two, the solutions sets of equations in free groups, normal forms for fundamental groups of 3-manifolds, or the words that represent non-trivial elements in the Grigorchuk group, are all examples of L systems.

In this talk I will give all the language definitions, and discuss as many of the examples above as time will allow.

Ewa Bieniecka, Daniel Bennett, and Feyisayo Olukoya
*University of St Andrews
***
**

**Ewa Bieniecka**

**Title:** Free products in R. Thompson’s group V

**Abstract:** Historically an approach to showing a group of permutations
factors as a free product of its subgroups is to show the existence of
“Ping-Pong” dynamics. However, it is the case that one can find permutation
groups which factor as free products but without Ping-Pong dynamics. In recent
years it has become a question as to whether any free product of subgroups of V
admits Ping Pong dynamics in its natural action on Cantor space. In this talk
we discuss some results related to this question. Joint work with Collin Bleak
and Francesco Matucci.

**Daniel Bennett**

**Title:** An introduction to a class of co-context free Thompson-like groups

**Abstract:**
In 2014 Witzel and Zaremsky introduced new Thompson-like groups based on the
Zappa-Szép product of monoids. It was subsequently shown by Berns-zieve, Fry,
Gillings, Hoganson and Mathews that a class of these groups, \(V_{aug}\), had the
property of being co-context free. We present a brief exploration of these
groups and our work involved in attempting to use the groups as counter
examples to Lehnert’s conjecture for V.

**Feyisayo Olukoya**

**Title:** The rational group and some of its subgroups

**Abstract:**
I will give a brief introduction to the rational group, highlight some of its
interesting subgroups and note along the way some results about these
subgroups.

John Gimbel
*University of Alaska
***A few parameters in fractional graph theory
**

Many branches of mathematics have seen so called fractional reinterpretations of their discipline. E.g. Fractional geometry and fractional calculus. This talk is meant as a gentle introduction to fractional graph theory. In doing so, we will consider several parameters--domination, coloring and cocoloring and their fractional counterparts.

Tomas Nilson
*Mid-Sweden University
***Agrawal’s conjecture for triple arrays
**

A triple array is an array in which two 2-designs are merged together such that any row and column contain the same number of common symbols.

Agrawal’s conjecture says that (in the canonical case), there is a triple array if and only if there is a symmetric 2-design (SBIBD).

Given a triple array we can construct a SBIBD. But here we will look at approaches and problems surrounding the desirable and still open direction. How to construct an array with these properties from a subset structure.

We will give background and define objects used. The exposition will be elementary and special knowledge of the area will not be assumed.

Robert Bailey
*Memorial University of Newfoundland
***Metric dimension, computation and distance-regular graphs
**