Algebra & Combinatorics Seminars

Previous Seminars - 2016 to 2017

Previous seminars from: 2018/19, 2017/18, 2016/17, 2015/16, 2014/15, 2013/14, 2012/13, 2011/12, 2010/11, 2008/09, 2007/08

Wednesday the 17th of May 2017 at 1pm in Theatre D

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.

Wednesday the 10th of May 2017 at 1pm in Theatre D

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.

Wednesday the 3rd of May 2017 at 1pm in Theatre D

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.

Wednesday the 26th of April 2017 at 1pm in Theatre D

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?

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.

Wednesday the 19th of April 2017 at 4pm in Lecture Theatre D

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.

Wednesday the 12th of April 2017 at 1pm in Theatre D

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.

Wednesday the 5th of April 2017 at 1pm in Lecture Theatre D

John Gimbel
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.

Wednesday the 8th of March 2017 at 1pm in Theatre D

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.

Friday the 24th of February 2017 at 2pm in Lecture Theatre A

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