The Pure Postgraduate Seminars are a series of talks given by and for the postgraduates in pure mathematics and CIRCA at the University of St Andrews. Each week a postgraduate will give a talk on a research subject of their choice - this can be on their current research, or anything else that might interest them. PPS aims to give postgraduates an opportunity to learn what their peers are working on, and to practice presenting their work in a relaxed environment.

All postgraduates are invited to speak at and attend the seminars. Any interested undergraduate students in senior honours are also welcome to attend. If you do not fit in any of those categories and would like to come along to PPS, please contact the organiser.

Seminars take place on Monday at 11am, in Lecture Theatre D. A weekly confirmation email will be sent round with details of the week's talk. If you would like to give a talk, please get in contact with me.

### Academic Year 2017/2018 (Semester 2):

Date Speaker Title & Abstract
8th February 2018 Nayab Khalid A New Normal Form for Thompson's Group F

Abstract:

Thompson's group F is a well-known example of a finitely presented infinite group. We discuss it as a rearrangement group of a fractal (Belk, Forrest 2016). We present a new generating set, and subsequently a new normal form for it - with the aim of generalizing this presentation for the wider class of rearrangement groups.
12th February 2018 Han Yu Luggage Packing

Abstract:

We discuss some packing problems of finite points configurations in Euclidean spaces. To give some ideas, let us consider a family of scaled copies of the 0-skeleton of the unit cube. We need to place the points in the Euclidean plane and minimize the 'space cost'. In this talk, I will give a precise formulation of such packing problems and sketch some proofs.
19th February 2018 Craig Miller Right noetherian semigroups

Abstract:

A semigroup S is right noetherian if every right congruence on S is finitely generated. In this talk, we will begin by looking at some known examples of right noetherian semigroups. We will then consider whether the property of being right noetherian is preserved by various semigroup constructions.
26th February 2018 Ashley Clayton Some congruence questions for subdirect products

Abstract:

In this talk, I will give a general overview of some upcoming results in the theory of subdirect products for semigroup and monoids, in particular highlighting some problems and questions for congruences of subdirect products and fiber products.
5th March 2018 Chris Russell Creating a database of 0-simple semigroups

Abstract:

0-simple semigroups are in some sense the building blocks of semigroups. Enumerating these semigroups up to isomorphism boils down to finding certain matrices up to a special equivalence relation. I will explain how I created a database of these for orders < 49, focusing on the strategies that allowed me to compute to higher orders.
12th March 2018 Raad Al Kohli Sufficient Conditions for Groups

No abstract was supplied
19th March 2018 Easter PPS An egg & rabbit themed quiz was held in place of a talk on this date

26th March 2018 Matt McDevitt Permutations in substructure setting

Abstract:

We consider permutations as sequences of integers and introduce a substructure relation on them. We then turn our attention to permutation anti-chains and explore some recent classification results.
2nd April 2018 Finn Smith TBA

9th April 2018 Mun See Chang TBA

16th April 2018 Euan Grant (Guest from Divinity) Divinity & Mathematics

23rd April 2018 Stuart Burrell TBA

### Academic Year 2017/2018 (Semester 1):

Date Speaker Title & Abstract
2nd October 2017 Fernando Flores Brito Congruences of rank 1 in EndF_n(G)

Abstract:

We will classify all of the congruences of rank one in EndF_n(G).
9th October 2017 Matt McDevitt Insertion Relations on Words

Abstract:

Using transducers, we introduce a class of relations on words which generalise the subword order, and discuss the decidability of their properties.
16th October 2017 Han Yu Fourier series and extra conditions of Duffin-Schaeffer conjecture

Abstract:

Duffin-Schaeffer conjecture concerns approximating real numbers by rationals in lowest form. It is an important topic in Diophantine approximation. Although no solution of this conjecture has been found, there are some partial results proving some weaker forms by strengthening some conditions in the statement of this conjecture. Here, in this talk, we shall see that we can use Fourier analysis to make a tiny step forward and show that the Duffin-Schaeffer conjecture holds under extra logarithmic divergence and logarithmic Vaaler type upper bound.
23rd October 2017 Lawrence Lee Diophantine approximation on manifolds and lower bounds for Hausdorff dimension

Abstract:

Diophantine approximation is a branch of analytic number theory that aims to understand how well real numbers may be approximated by rationals. In this talk I will provide an introduction to Diophantine approximation for those who are unfamiliar with the subject. I will also show how we can establish analogues of classical theorems from the real line on more complicated spaces, such as manifolds.
30th October 2017 Craig Miller Finiteness properties of M-acts

A record for this talk is unavailable.
6th November 2017 Ashley Clayton Fiber products of free objects

Abstract:

We consider finiteness properties for fiber products of free monoids and free semigroups, and ask some related questions.
13th November 2017 (1) Finn Smith Baire Category and Box Dimension

Abstract:

In the 1960s Gruber showed that typical (in the sense of Baire category) compact subsets of a metric space are badly-behaved with regard to box-counting dimension. In this talk I will introduce these notions, then discuss Gruber's result and how it may be generalised.
13th November 2017 (2) Nayab Khalid A New Normal Form for Thompson's Group F

Abstract:

Thompson's group F is a well-known example of a finitely presented infinite group. We discuss it as a rearrangement group of a fractal (Belk, Forrest 2016). We present a new generating set, and subsequently a new normal form, for it - with the aim of generalizing this presentation for the wider class of rearrangement groups.
20th November 2017 Douglas Howroyd Sum Fractals

An abstract was not provided for this talk.
27th November 2017 Gerard O'Reilly The Word Problem for Inverse Monoid Presentations.

Abstract:

We will explore the theory and techniques developed by Stephen that allow the word problem for certain classes of inverse monoid presentations to be solved through connections with formal language theory.
4th December 2017 Wilf Wilson Computing direct products of semigroups

Abstract:

The direct product of semigroups is easy to define and imagine in the abstract. But given two (or more) semigroups, on a computer, say, how do can go about actually getting your hands on the direct product: what representation can you use, and what generating set? I’ll talk about these questions, and some related ones.
11th December 2017 Christmas PPS (Michael Torpey) Christmas Quiz

Mince pies were provided with a christmas quiz, and a good time was had by all.

### Academic Year 2016/2017 (Semester 2):

Date (& room) Speaker Title & Abstract
6th February 2017,
Theatre A
Han Yu An introduction to additive combinatorics

Abstract:

Additive combinatorics is addictive. It is a very alive topic nowadays and has a lot of applications in number theory, harmonic analysis and fractal geometry. We will see some simple results from additive combinatorics and have fun with sum sets.
13th February 2017
Theatre D
Michael Torpey An introduction to GAP

Abstract:

Computational algebra is a popular field of study in St Andrews, and the go-to computer algebra package is GAP. In this talk we will introduce GAP and its packages, give a few demonstrations of what can be done with it, including computing the permutations of a Rubik's cube, and show off some upcoming features of the Semigroups package. Anyone wishing to "play along" should bring a laptop along, after installing GAP from here: http://www.gap-system.org/Releases/index.html
20th February 2017,
Theatre A
Fernando Flores Brito The Principal Factors of The Full Transformation Monoid, The Structure of its D-Classes And Similarities With The Semigroup Wreath Product

Abstract:

We will study the structure of the D-classes of the principal factors of the full transformation monoid on a finite set. In doing so, hopefully we will appreciate how different R-classes and different L-classes within the same D-class have different shape in terms of the location of their H-Group classes and null semigroups. This will allow us to use similar techniques when studying the D-classes of the Wreath product of the full transformation monoid and a permutation group, when considered as the endomorphism semigroup of free-G acts
27th February 2017,
Theatre D
Nayab Khalid Rearrangement Groups via Wreath Products

Abstract:

A quick revision of the (generalized) wreath product, and a discussion on how it can be used to determine if a rearrangement group is finitely generated.
6th March 2017,
Theatre A
Wilf Wilson The maximal subsemigroups of some special monoids.

Abstract:

At previous postgraduate pure seminars, I have described how to compute the maximal subsemigroups of an arbitrary finite semigroup. In this talk, I will apply these techniques to some examples of monoids of order/orientation-preserving/reversing partial transformations. Much of the work is involved in counting and describing the maximal independent subsets of particular bipartite graphs. This involves some nice combinatorics. The Fibonacci numbers will appear!
27th March 2017,
Theatre D
Craig Miller Large Subacts

Abstract:

This talk will first introduce the theory of M-acts and then we will discuss large subacts of M-acts. If B is a subact of an M-act A with A\B is finite, then B is called a large subact of A and A is called a small extension of B. In this talk we will investigate certain properties, such as finite generation and finite presentability, to see if they are preserved by large subacts and small extensions.
3rd April 2017,
Tutorial Room 3B
Douglas Howroyd Dimension and regularity of measures

Abstract:

Many of the techniques used to calculate the dimensions of a fractal rely on the properties of underlying measures. This had lest to the study of the dimension of a measure and how these definitions all interact. We will introduce a new definition of a measure and will discuss some of its properties, notably the overlap with the doubling property and how these results help in the study of weak tangent measures. This is joint work with Jonathan Fraser.
10th April 2017,
Theatre D
Dan Bennett To be a universal coCF group or not to be a universal coCF group, that is the question (that I sadly fail to answer in my thesis).

Abstract:

Let G be a group with presentation . The word problem of G is the algorithmic problem of determining, in a finite number of steps, whether a finite product of generators from X will be equal to the identity of G. In other words, finding a finite process that "identifies" the set of strings WP(G):={w|w=1, w in X*} and rejects the rest. Additionally there is also the co-word problem of G which asks the same question of the set coWP(G):={w|w not the identity}. In 1971 Anisimov proved that WP(G) is a regular language if and only if G is finite. Further, in 1983-85 Muller and Schupp proved that the WP(G) is a context free language if and only if G is a finitely generated virtually free group. The next natural question to ask is whether the groups whose co-word problem is context free (coCF groups) also have a nice classification? A version of Lehnert's conjecture states that Thompson's group V is a universal coCF group, i.e. contains copies of all coCF groups as subgroups . My thesis investigates this conjecture by exploring potential counter examples to the claim. Unfortunately none of the work has provided an answer one way or another, however in the talk I will take you through the journey I've taken and some results that have been found.
17th April 2017,
Theatre A
Tom Bourne English, French, Spanish... A Variety of Languages.

Abstract:

In certain situations, it is easier to consider a collection of regular languages as a whole rather than as individuals. In this talk, we will encounter varieties of languages and their counterpart, pseudovarieties of monoids, and discuss how (not) to form them. Examples galore!
24th April 2017,
Theatre D
Shayo Olukoya Through the looking glass: some reflections on a couple of rabbit holes I have encountered.

Abstract:

I will talk about a couple of problems I started thinking about for fun and how they ended up having nice consequences.

### Academic Year 2016/2017 (Semester 1):

Date Speaker Title & Abstract
29 September 2016 Ashley Clayton Structures of inverse semigroups arising from partial mappings

Abstract:

Let $$X$$ be a set. A partial mapping of $$X$$ is an assignment of elements from $$A$$ to $$B$$, where $$A$$ and $$B$$ are some subsets of $$X$$. Under the correct composition of partial mappings, the set of all injective partial mappings on $$X$$ forms a semigroup known as the symmetric inverse semigroup, denoted $$\mathcal{I}_{X}$$. We naturally ask questions relating to the structure of $$\mathcal{I}_{X}$$ for finite and infinite $$X$$; including classifying Green's relations on $$\mathcal{I}_{X}$$, embedding properties, maximal subgroups and congruences, and generalisations of standard results of group theory to inverse semigroup theory. In particular, we show exactly when $$\mathcal{I}_{X}$$ is fundamental and factorisable as a semigroup, as well as considering two important submonoids of $$\mathcal{I}_{X}$$ which relate to the study of independence algebras and reflection monoids respectively. The questions of structure are also asked and subsequently answered for these examples, and we discuss their applications to the fields of fermion and boson annihilation, solid state physics model theory, $$C^{*}$$ algebras and the decidability of the generalised word problem in computer science.

06 October 2016 Matt McDevitt Subword-like orders

Abstract:
We discuss orders on words and consider generating subword-like relations using transducers.

13 October 2016 Han Yu On generalized trigonometric functions and series of rational functions

Abstract:
It is well known that we can express infinite series of rational functions with polygamma functions by using partial fraction decomposition, here we consider the two sided sum $$\lim_{k\to\infty}\sum_{n=-k}^{k}f(n)$$ where $$f$$ is any rational function such that limit is finite. In this special case, we can compute the series explicitly without using polygamma functions which in turn gives us some more information about polygamma functions. To do this we introduced a class of generalized trigonometric functions which include trigonometric functions as special cases and share some similarities such as summation formula and algebraic intentities.

20 October 2016 Sascha Troscheit Stochastic processes and trees using linear approximations

Abstract:
In this talk we will be looking at collections of trees, forests if you will, and stochastic processes associated with them. We will look at several branching processes: random homogeneous, random recursive, and a model called $$V$$-variable. The latter was introduced to interpolate between random homogeneous (also called $$1$$-variable) and random recursive (also known as $$\infty$$-variable) and we will try and shed some light on whether it actually does. Most of the talk will focus on somewhat challenging open problems and I will present my progress and realisations to date. (Mostly using "obvious" approximations to the processes)

27 October 2016 Nayab Khalid Generalized Thompson's Groups

Abstract:
The aim of this talk is to introduce the audience to Richard Thompson's Groups $$F$$,$$T$$ and $$V$$, as well as some generalizations of these groups. The infinite family of groups $$nV$$ is of special interest. We will define these groups and discuss some properties, and time permitting, we will present Brin's proof of simplicity of $$nV$$.

03 November 2016 Douglas Howroyd Fractals

Abstract:
Fractals are often thought of as objects with some geometric detail that manifests at all scales. As such the study of these objects requires looking at infinitesimal properties of sets. One of the principal ways of doing this is to find the Hausdorff dimension of the object, but for many fractals this calculation is quite difficult when working from first principles. We will provide a brief introduction to fractal geometry starting with the definition of Hausdorff dimension. Then we introduce iterated function systems and explain how they can simplify many problems. Finally we will finish by explaining a famous result by Falconer on the dimension of self-affine sets.

10 November 2016 Feyisayo Olukoya Subgroups and overgroups of $$V_{n}$$ via topological conjugation

Abstract:
See title. Joint work with Casey Donoven.

17 November 2016 Chris Russell Primitive Permutation Groups

Abstract:
This talk will introduce primitive permutation groups and attempt to justify the statement 'primitive permutation groups are the building blocks of all permutation groups'.

24 November 2016 Michael Torpey Diagram Semigroups

Abstract:
You've studied permutations to death. You've probably even looked at transformations, and partial permutations. But what if we relaxed some more definitions? How much more general could we get? In this adventure through combinatorics and semigroup theory, we will start at permutation groups, then keep generalising and generalising until there's nothing left to generalise and so little structure left that you'd hardly believe it's still worth studying. A fun interactive worksheet will be provided to help you experience the weightless feeling of PBRs and the elegant beauty of Temperley-Lieb diagrams for yourself. Bring a pen!
(slides and worksheet)

01 December 2016 Tom Bourne Subwords and Stars

Abstract:
In the field of formal language theory, the generalised star-height problem asks whether or not there exists an algorithm to determine the minimum nesting depth of stars required in order to represent a given regular language by a regular expression. In this setup, we consider complement as a basic operator. In particular, it is not yet known whether there exist languages of generalised star-height greater than one. This talk will consider the generalised star-height of the languages in which a fixed word occurs as a contiguous subword an exact number of times and of the languages in which a fixed word occurs as a contiguous subword modulo a fixed number, and prove that in each case it is at most one.

08 December 2016 Craig Miller Presentations of $$M$$-acts

Abstract:
This talk will first introduce the theory of $$M$$-acts and then we will discuss presentations of acts. We will consider monoids $$M$$ for which the class of finitely generated $$M$$-acts coincides with the class of finitely presented $$M$$-acts. In particular, we will show that right noetherian monoids have this property.

15 December 2016 Christmas PPS Bring along a christmas related maths fact/problem/theorem and we'll have a maths-christmas party!

### Academic Year 2015/2016 (Semester 2):

Date Speaker Title & Abstract
10 February 2016
Wilfred Wilson
Computing maximal subsemigroups: the conclusion

Abstract:
Previously, You may have heard me speak about finding the maximal subsemigroups of some specific types of semigroup, in particular Rees 0-matrix semigroups - and perhaps I mentioned some horrible ways to use that information to calculate the maximal subsemigroups of an arbitrary semigroup.
After much thought, it turns out that those ways don't need to be anywhere near as horrible as I once feared. In this talk, I'll talk about the recent advances that we've made towards computing the maximal subsemigroups of an arbitrary finite semigroup, and these advances largely put this issue to rest (for now).

17 February 2016
Rachael Carey
Act natural! Disjoint unions of the free monogenic semigroup.

Abstract:
We examine the restrictions on multiplication in semigroups which are disjoint unions of finitely many copies of the free monogenic semigroup (as described by Abu-Ghazalh and Ruskuc), and see how all such semigroups are unary graph automatic.

9 March 2016
Casey Donoven
Invariant Relations and Edge Replacement Systems

Abstract:
I will speak about two types of relations on Cantor space, invariant relations and gluing relations. After studying invariant relations for some time, Martyn and Collin were worried that Nayab's work with edge replacement systems and the resulting gluing relations were the same as my work. I will show you why they thought this and how they were wrong.

23 March 2016
Matt McDevitt
Well-quasi-ordering the free monoid

Abstract:
A well known result of Higman shows that the free monoid is well-quasi-ordered (wqo) under the (non-contiguous) subword order. By contrast, it isn't wqo under the (contiguous) factor order. We discuss an ongoing research project into some orders which fall between the subword and factor orders.

30 March 2016
Nayab Khalid

Abstract:
Richard Thompson's groups are only one example of rearrangement groups of fractals. I will be continuing my discussion of the larger class of rearrangement groups which act on fractal spaces with a graph-like structure.

6 April 2016
Michael Torpey
Dolphin Semigroups

Abstract:
We spend a lot of time considering semigroups of transformations, words, permutations and so on, but a semigroup can in fact contain any type of object. In this talk I will consider a few different ways to multiply dolphins and, if time allows, other cetaceans. No prior knowledge of marine biology is assumed.
slides

13 April 2016
Sascha Troscheit
Measures, gauges and iterating the logarithm

Abstract:
Be afraid. Be very afraid! This talk is all about random variables, expected values, and the most likely worst case scenario. I will explain what the law of the iterated logarithm is, how this ties in with the Hausdorff measure for random self-similar sets (it's zero), and what we can do to force a positive and finite measure, almost surely. Simply put: the Hausdorff measure is a poor choice and there are better choices for a dimensional measure.
Bring a coin to participate.

20 April 2016
Tom Bourne
You want me to count the arrows? I'll $$C_{2}$$ it!

Abstract:
Given a complete deterministic finite state automaton over an alphabet $$A$$, pick an arrow, say $$(q,a)$$, where $$q$$ is a state in the automaton and $$a$$ is a letter in $$A$$. What is the generalised star-height of the language that counts travelling through this arrow $$k$$ modulo $$n$$ times?
With certain automata this question corresponds to the generalised star-height of languages recognised by wreath products of commutative and cyclic groups. We'll see what happens in the case where the cyclic group is $$C_{2}$$ and why Colva and I have hit a brick wall when we try the same idea with $$C_{3}$$ instead.

27 April 2016
Daniel Bennett
Fruit trees? Creating presentations for Thompson-like groups.

Abstract:
Yes (they're not actual fruit trees)

### Academic Year 2015/2016 (Semester 1):

Date Speaker Title & Abstract
2 October 2015
Michael Torpey
Semilattice Congruences

Abstract:
A semilattice is a partially ordered set such that every subset has a greatest lower bound. It turns out that a semilattice forms a semigroup under the operation of taking greatest lower bounds, and such a semigroup has a number of interesting properties. In this talk, over cake, we will consider the congruences on a semilattice, and discuss how we could compute efficiently with them.
slides

9 October 2015
Casey Donoven
Transducers, Groups, and Applications

Abstract:
I will describe what transducers are, a few groups associated with transducers, and a real-life application!

16 October 2015
Nayab Khalid
Tensor Products (via the Universal Property)

Abstract:
In this talk we will define the tensor product using the universal property. We will also prove, using this property, that the tensor product is uniquely defined, up to isomorphism.
slides

23 October 2015
Ewa Bieniecka
Dynamical properties in Thompson's group V

Abstract:
In this talk we will develop methods for describing dynamical properties of elements of Thompson's group V, as well as discuss problems and constructions in which using those properties play a central rôle.

30 October 2015
Davide Ravotti
(University of Bristol)
Time-changes of Homogeneous flows

Abstract:
Homogeneous Dynamics is a branch of Ergodic Theory which has been studied for many years and has shown to have deep and surprising connections with Number Theory; roughly speaking, it deals with flows on (quotients of) Lie groups. We will discuss properties of time-changes of these flows, namely what happens when you keep the same orbits but you change the speed of the points.
slides

6 November 2015
Julius Jonusas
Universal Talk

Abstract:
$$\varnothing$$

13 November 2015
Matt McDevitt
Complete Mappings of Finite Groups

Abstract:
The multiplication table of a finite group can be considered as a Latin square. We investigate which groups' multiplication tables have an orthogonal mate, via a characterisation using complete mappings.

20 November 2015
Tom Bourne
Conversations on Counting Kleene Stars

Abstract:
The generalised star-height problem is one of many questions in the field of formal language theory that is easy to state but extremely difficult to answer. In this talk we'll explore how combinatorics on words might be one of the better approaches to take when tackling the problem and see how this approaches helps us to calculate the star-height of languages recognised by Rees matrix semigroups over commutative groups.

27 November 2015
Wilf Wilson
How does the Semigroups package work?

Abstract:
It's taken me years, but I've finally got around to learning (roughly) how the Semigroups package works. If you�ve ever wondered how the Semigroups package can quickly calculate things about certain types of semigroup, without needing to get all the elements, then this talk is for you. I'll describe the fundamental ideas, then run through many examples by hand.

4 December 2015
Feyishayo Olukoya
Dynamics and Decision Problems in $$V$$

Abstract:
We shall consider how the dynamical properties of $$V$$ help in answering decision problems. If time permits we shall present a solution to the conjugacy problem in $$V$$.

11 December 2015
Daniel Bennett
More dynamical stuff in Thompson's group $$V$$

Abstract:
Thompson's group $$V$$ contains a interesting set of subgroups that are defined by a certain dynamical property. Any group isomorphic to one of these special subgroups is called a "demonstrable" group with respect to $$V$$. It has been recently shown that the finitely generated demonstrable groups are exactly the set of context free groups. In the talk we will explore what all these terms mean and time permitting show one side of the theorem.

### Academic Year 2014/2015 (Semester 2):

Date Speaker Title & Abstract
10 February 2015
Michael Torpey
DiSparse6: a handy way for computers to remember digraphs

Abstract:
Are you sick of drawing huge, complicated diagrams to represent directed graphs? Sick of laboriously typing out huge lists of vertices and edges in emails and text messages to share interesting digraphs with your friends and colleagues? There is another way! In this talk, we present the brand new DiSparse6 format to condense arbitrary digraphs to short, computer-readable, printable ASCII strings. We explain the format, the algorithm, and the complexity of the resulting string, and give you a chance to join in and make your very own DiSparse6 string to take home with you. Bring a pen!

slides

17 February 2015
Arthur Schaefer
On Princesses and Decompositions

Abstract:
Synchronization theory is the theory of full transformation semigroups containing constant functions. The recent approach on this theory uses permutation groups and their symmetry to determine whether or not a semigroup S of the form $$S=\langle g,t\rangle$$ is synchronizing ($$G$$ the group of units of $$S$$ and $$t$$ a transformation). This method has led to many ground breaking insights on semigroups. In this talk I will demonstrate that some non-synchronizing semigroups have an interesting decomposition property which I will describe as follows. If $$S$$ is a semigroup, then there are subsemigroups $$S_1$$ and $$S_2$$ such that $$S$$ is the disjoint union of its subsemigroups. There is little theory on such decompositions of semigroups; therefore, this talk will mainly contain developments which led me to this kind of decompositions and examples, including the current research.

slides

24 February 2015
Julius Jonusas
Computers

Abstract:
Schreier-Sims

3 March 2015
Rachael Carey
Say aaaaaa: Unary Graph Automatic Semigroups

Abstract:
A graph automatic semigroup can be represented by a regular language. What happens if we restrict ourselves to languages over a single letter alphabet? Come to PPS to find out.

slides

10 March 2015
Sascha Troscheit
Logic, typos and comics

Abstract:
As I am consistently being accused of not being a "pure mathematician" I thought I would talk about something somewhat more abstract than abstract shapes. So this talk will encompass abstract logic, axioms, inconsistency, consistency, embarrassing typos and comics.
This is either a serious or not a serious talk.
And this is definitely a lie.

31 March 2015
Wilfred Wilson
Transformation semigroups and minimal ideals

Abstract:
Transformations are to semigroups as permutations are to groups, and I�ve been thinking about them a lot recently. I�ll share my thoughts, and talk about some ways of efficiently calculating properties of a transformation semigroup related to its minimal ideal.

slides

6 April 2015
Jasmina Angelevska
(Ss. Cyril and Methodius University of Skopje)
Something old, something new, something borrowed.

14 April 2015
Feyisayo Olukoya
Homotopy, Homology, and the fundamental group.

Abstract:
I will give a survey of one of the topics I enjoyed most over the course of SMSTC.

21 April 2015
Daniel Bennett
Not Seeing the Forest for the Trees

Abstract: Taking a step back from binary trees, we will consider the larger Forest Monoid and how to use it to create Thompson-like groups. At it�s core is the Zappa-Sz�p product between monoids which Matt Brin took and applied to the specific case of the Forest Monoid to create the BZS-product. We will give a hand-wavy explanation of how Thompson�s group V can be derived from such a construction, and finish by presenting some more interesting groups that arise in the same fashion.

28 April 2015
Jose Espin Buendia
(University of Murcia)
(Published) Mistakes which motivate talks, books and even PhD theses!

Abstract:
In The International Congress of Mathematicians in 1900, Hilbert posed a list of 23 problems. In the second part of the 16th problem (still open) he proposed to find (if possible) an upper bound for the number of limit cycles that a planar polynomial vector fields of degree $$n$$ can have.
In 1923, H. Dulac presented a proof of a partial result: any planar polynomial vector field have finitely many limit cycles.
Since the publication of this result by Dulac, many results in the field of qualitative theory of differential equations have been proved using it (in a essential way). However, in 1982, Yu.S. Ilyashenko found a fatal mistake in Dulac's proof. Ilyashenko itself was able to amended the proof (the proposed new proof is a book!) in 1991. J. Ecale (in 1992), found a different proof as well. Despise these two proofs are generally approved by the mathematical community, they are extremely difficult to follow even to specialists in the field.
The research I am doing as a PhD student is motivated by a (modern) situation similar to the one described above. In 2007, V. Jimenez (my advisor) and J. Llibre gave a topological characterizations of the omega-limit sets for analytic flows on the sphere and the projective plane. Their proof is based on an auxiliary lemma which, despite the validity of its statement, it is not well proved in the paper. I shall present how we found the problem in that paper (just the context, I will not bore you with technical details!) and (the idea of) how we solved it.
I would like to take advantage of this talk to propose a discussion about what proving in maths really is.

4 June 2015
Hendrik Weyer
(University of Bremen)
Spectral properties of measure-geometric Laplacians

Abstract:
In this talk we will discuss measure-geometric Laplacians $$\Delta^{\mu}$$ with respect to compactly supported atomless Borel probability measures $$\mu$$ as introduced by Freiberg and Z�hle in 2002. We will recall various basic properties and present recent results on the spectral properties of $$\Delta^{\mu}$$. Particularly we will classify the eigenfunctions of $$\Delta^{\mu}$$ under both Dirichlet and von Neumann boundary conditions. We will conclude by illustrating our results through several examples.
(This is joint work with M. Kesseb�hmer and T. Samuel)

### Academic Year 2014/2015 (Semester 1):

Date Speaker Title & Abstract
3 October 2014
Sascha Troscheit

Abstract:
In this talk I will give a short introduction to Assouad dimension and how it differs to other types of fractal dimension�. The main part of the talk will consist of introducing various models of random fractals and a brief survey of work done in the area. The talk will involve lots of handwaving and plenty of pictures and very little actual analysis.
(Joint work with Jonathan Fraser and Jun Jie Miao)
slides

10 October 2014 Tom Bourne
Multilingual Monoids: Why Study Varieties of Languages?

Abstract:
An introductory talk into why we should study the connections between families of monoids and families of languages, rather than individual monoids and individual languages. Variety theory will be introduced in order to highlight Eilenberg's Variety Theorem as the essential part of the framework. Examples throughout (hopefully)!
slides

17 October 2014
Artur Sch�fer
The central dogma of genetics: Or rather the coding theory behind it

Abstract:
The central dogma of genetics says, that the coded genetic information (DNA) is transcribed into transportable cassettes, composed of messenger RNA which contains the program for synthesis of a particular protein. In short: Information is encoded, sent through a channel and decoded into useful information. Having this highly interesting application as our background, we are going to take a look at the basics and non-basics of coding theory. Codes can be defined using as little algebraic structure as possible, but by providing more it might generate better codes. Thus, we will start from looking at basic codes, moving to linear and cyclic codes and afterwards presenting group ring codes.
Following that approach, the "amount" of algebraic structure is increasing, but does it always give us better codes?
And what are good and bad codes, actually?
slides

24 October 2014
Casey Donoven
Friendly Introduction to Fractal Dimension

Abstract:
I will introduction the Hausdorff dimension and give some motivation for its existence through examples and less intricate notions of dimension.  I might also describe the box-counting dimension and some multifractal theory.

31 October 2014
Michael Torpey
Computing with Semigroup Congruences

Abstract:
Congruences crop up everywhere in algebra, though we may not always know it. Any algebraic object has attached to it some theory which allows us to
partition it, take quotients and find homomorphic images.  In group theory we
use normal subgroups; in ring theory, two-sided ideals. In semigroup theory
in general, we can't do much better than study congruences directly, which is
very expensive if we want to do anything computational. However, after some
definitions, I will present a few different ways that we can abstract
semigroup congruences to other objects, and we will see how these abstractions
might help us compute things we want to know.

7 November 2014
Wilfred Wilson
Computing the maximal subsemigroups of finite semigroups

Abstract:
It is easy to ask (but much more difficult to answer) the question: given a
finite semigroup, what are its maximal subsemigroups? Fortunately there
is enough theory to allow us to compute these things in a practical way.
We firstly calculate the maximal subsemigroups of a few easy classes of
semigroup, and then proceed to discuss how the "MaximalSubsemigroups"
function works in the Semigroups package.

14 November 2014
Julius Jonusas
Approximations in automorphism groups

Abstract:
Short introduction to Fra�ss� limits and approximations of automorphisms on them.

21 November 2014
Feyisayo Olukoya
The conjugacy problem in groups

Abstract:
The conjuagacy (or transformation) problem is one of three group-theoretic decision problems proposed by Max Dehn in 1912 which he considered to be fundamental to group theory. The conjugacy problem has since been proven to be undecidable in general. There has been a lot of work done on the conjugacy problem, too much to be covered in the scope of our discussion and with many remarkable results, hence we shall be focusing only on some of these results. We begin by introducing the problem and considering a solution to the transformation problem in finite groups --- firstly the finite symmetric group and then in greater generality all finite groups.The finitely generated free groups will give us an example of an infinite group with solvable conjugacy problem. We then move on to look at insolubility cases, which will naturally force a more thorough examination of the definitions of algorithm, decidability, undecidability, semi-decidability solvability and insolubility, and how these arise in infinite groups. As a result of this, we are led to examine the question of the interplay between the conjugacy problem and certain familiar group-theoretic constructions. In this we shall be focused on extensions of groups and here we shall present the result which will form the highlight of the discussion time permitting.

This talk is based entirely on my final year project.

28 November 2014
Daniel Bennett
Demonstrative Embedding's in Thompson's Group V

Abstract:
There exists a conjecture in the area of CoCF groups, that was proposed in it's original form by Lehnert and revised by Bleak, Matucci and Neunhoffer, which proposes that Thompson's Group V is a Universal CoCF Group. We will explore what this means and steps that are being taken to prove/disprove this conjecture. An active area of study that looks promising in this regard are the demonstrative subgroups of V. The second half of the talk will be uncovering what these are, looking at the closure properties of these groups and ending with their relevance to the current conjecture.

5 December 2014
Gavin Abernethy
Normal numbers

Abstract:
I shall be discussing some ideas about normal and non-normal numbers, using results regarding the uniform distribution of sequences as well as from ergodic theory. This talk shall be based on a part of my senior honours project from last year.

12 December 2014
Christmas Lunch

### Academic Year 2013/2014 (Semester 2):

Date Speaker Title & Abstract
4 March 2014
Thomas Bourne
Minimal Automata and the Method of Quotients

Abstract:
We will explore how to algorithmically construct the minimal automaton for a regular language given some regular expression representing said language. The technique used to do this is the Method of Quotients. The theory will be demonstrated with an extended example.
11 March 2014
Abel Farkas
Projections and other images of self similar sets

Abstract:
It will be measure theory and dimension theory of mainly linear images of self-similar sets.
8 April 2014
Casey Donoven
Intersections within Cantor Spaces.

Abstract:
The codimension formula describes the dimension of the intersection of two fractal in R^n. I will explain my recent work with Kenneth on abstracting this result in Cantor Spaces.
15 April 2014
Sascha Troscheit
Recurrence and the renewal shift.

Abstract: In this talk I will describe the notion of recurrence in dynamical systems and give a combinatorical approach to the analysis of subsets of the shift space associated with the renewal shift with specified recurrence rates.
22 April 2014
Jenni Awang
Sorry we're closed, but that's a nice dress (aka Closures and Compl{i,e}ments)

Abstract:  A 1922 paper by Kuratowski described what happens when you take repeatedly take closures and complements of a set in a topological space.  We'll look at his result, and then swiftly move away from the world of topology and see how this can be applied to formal languages (Brzozowski, Grant, Shallit, 2009).
29 April 2014
Rachael Carey
Recognising semigroups

Abstract: Automatic semigroups, graph automatic semigroups and FA-presentable semigroups are all types of semigroups which can be represented by regular languages and recognised by finite state automata. In my talk I will define these concepts, outline some of the properties of each, and explain how they interact with one another.
6 May 2014
Artur Sch�fer
Synchronization and Graphendomorphisms

Abstract: In this talk we will introduce the term synchronizing permutation groups and find endomorphisms of graphs corresponding to non-synchronizing permutation groups. Only primitive groups can be synchronizing; therefore, we are able to apply the classification of O'Nan Scott of primitive groups and try to classify synchronizing groups. Also, groups can be attached to undirected graphs and the synchronization property can be translated into a graph property. Vaguely the theorem states: A group is synchronizing iff its graphs have no proper endomorphisms. We will apply this theorem to the Hamming graph and investigate the structure of its endomorphisms.
13 May 2014
Daniel Bennett
Context Free Grammars, Languages and Groups

Abstract: A brief introduction to Context Free Grammars and Languages. We will take the natural progression onto Context Free and CoCF groups, with examples of all these concepts throughout.
20 May 2014
Vuksan Mijovic Multifractal zeta functions

Abstract: Talk will be based on joint paper with Lars Olsen in which we introduce multifractal zeta-functions providing precise information of a very general class of multifractal spectra, including, for example, the multifractal spectra of self-conformal measures and the multifractal spectra of ergodic Birkhoff averages of continuous functions. More precisely, we prove that these and more general multifractal spectra equal the abscissae of convergence of the associated zeta functions.
27 May 2014

### Academic Year 2013/2014 (Semester 1):

Date Speaker Title & Abstract
10 October 2013
Casey Donoven
Groups of Homoeomorphisms of Cantor Sets

Abstract:
Self-similar Cantor sets allow for many interesting homeomorphism through exploiting their similar, disconnected, and symmetrical cylinders. I will discuss groups of such homeomorphisms, and through orbit dynamics and automata, distinguish an isomorphism class.
20 November 2015
Tom Bourne
Conversations on Counting Kleene Stars

Abstract:
The generalised star-height problem is one of many questions in the field of formal language theory that is easy to state but extremely difficult to answer. In this talk we'll explore how combinatorics on words might be one of the better approaches to take when tackling the problem and see how this approaches helps us to calculate the star-height of languages recognised by Rees matrix semigroups over commutative groups.

27 November 2015
Wilf Wilson
How does the Semigroups package work?

Abstract:
It's taken me years, but I've finally got around to learning (roughly) how the Semigroups package works. If you�ve ever wondered how the Semigroups package can quickly calculate things about certain types of semigroup, without needing to get all the elements, then this talk is for you. I'll describe the fundamental ideas, then run through many examples by hand.

4 December 2015
Feyishayo Olukoya
Dynamics and Decision Problems in $$V$$

Abstract:
We shall consider how the dynamical properties of $$V$$ help in answering decision problems. If time permits we shall present a solution to the conjugacy problem in $$V$$.

11 December 2015
Daniel Bennett
More dynamical stuff in Thompson's group $$V$$

Abstract:
Thompson's group $$V$$ contains a interesting set of subgroups that are defined by a certain dynamical property. Any group isomorphic to one of these special subgroups is called a "demonstrable" group with respect to $$V$$. It has been recently shown that the finitely generated demonstrable groups are exactly the set of context free groups. In the talk we will explore what all these terms mean and time permitting show one side of the theorem.

### Academic Year 2014/2015 (Semester 2):

Date Speaker Title & Abstract
10 February 2015
Michael Torpey
DiSparse6: a handy way for computers to remember digraphs

Abstract:
Are you sick of drawing huge, complicated diagrams to represent directed graphs? Sick of laboriously typing out huge lists of vertices and edges in emails and text messages to share interesting digraphs with your friends and colleagues? There is another way! In this talk, we present the brand new DiSparse6 format to condense arbitrary digraphs to short, computer-readable, printable ASCII strings. We explain the format, the algorithm, and the complexity of the resulting string, and give you a chance to join in and make your very own DiSparse6 string to take home with you. Bring a pen!

slides

17 February 2015
Arthur Schaefer
On Princesses and Decompositions

Abstract:
Synchronization theory is the theory of full transformation semigroups containing constant functions. The recent approach on this theory uses permutation groups and their symmetry to determine whether or not a semigroup S of the form $$S=\langle g,t\rangle$$ is synchronizing ($$G$$ the group of units of $$S$$ and $$t$$ a transformation). This method has led to many ground breaking insights on semigroups. In this talk I will demonstrate that some non-synchronizing semigroups have an interesting decomposition property which I will describe as follows. If $$S$$ is a semigroup, then there are subsemigroups $$S_1$$ and $$S_2$$ such that $$S$$ is the disjoint union of its subsemigroups. There is little theory on such decompositions of semigroups; therefore, this talk will mainly contain developments which led me to this kind of decompositions and examples, including the current research.

slides

24 February 2015
Julius Jonusas
Computers

Abstract:
Schreier-Sims

3 March 2015
Rachael Carey
Say aaaaaa: Unary Graph Automatic Semigroups

Abstract:
A graph automatic semigroup can be represented by a regular language. What happens if we restrict ourselves to languages over a single letter alphabet? Come to PPS to find out.

slides

10 March 2015
Sascha Troscheit
Logic, typos and comics

Abstract:
As I am consistently being accused of not being a "pure mathematician" I thought I would talk about something somewhat more abstract than abstract shapes. So this talk will encompass abstract logic, axioms, inconsistency, consistency, embarrassing typos and comics.
This is either a serious or not a serious talk.
And this is definitely a lie.

31 March 2015
Wilfred Wilson
Transformation semigroups and minimal ideals

Abstract:
Transformations are to semigroups as permutations are to groups, and I�ve been thinking about them a lot recently. I�ll share my thoughts, and talk about some ways of efficiently calculating properties of a transformation semigroup related to its minimal ideal.

slides

6 April 2015
Jasmina Angelevska
(Ss. Cyril and Methodius University of Skopje)
Something old, something new, something borrowed.

14 April 2015
Feyisayo Olukoya
Homotopy, Homology, and the fundamental group.

Abstract:
I will give a survey of one of the topics I enjoyed most over the course of SMSTC.

21 April 2015
Daniel Bennett
Not Seeing the Forest for the Trees

Abstract: Taking a step back from binary trees, we will consider the larger Forest Monoid and how to use it to create Thompson-like groups. At it�s core is the Zappa-Sz�p product between monoids which Matt Brin took and applied to the specific case of the Forest Monoid to create the BZS-product. We will give a hand-wavy explanation of how Thompson�s group V can be derived from such a construction, and finish by presenting some more interesting groups that arise in the same fashion.

28 April 2015
Jose Espin Buendia
(University of Murcia)
(Published) Mistakes which motivate talks, books and even PhD theses!

Abstract:
In The International Congress of Mathematicians in 1900, Hilbert posed a list of 23 problems. In the second part of the 16th problem (still open) he proposed to find (if possible) an upper bound for the number of limit cycles that a planar polynomial vector fields of degree $$n$$ can have.
In 1923, H. Dulac presented a proof of a partial result: any planar polynomial vector field have finitely many limit cycles.
Since the publication of this result by Dulac, many results in the field of qualitative theory of differential equations have been proved using it (in a essential way). However, in 1982, Yu.S. Ilyashenko found a fatal mistake in Dulac's proof. Ilyashenko itself was able to amended the proof (the proposed new proof is a book!) in 1991. J. Ecale (in 1992), found a different proof as well. Despise these two proofs are generally approved by the mathematical community, they are extremely difficult to follow even to specialists in the field.
The research I am doing as a PhD student is motivated by a (modern) situation similar to the one described above. In 2007, V. Jimenez (my advisor) and J. Llibre gave a topological characterizations of the omega-limit sets for analytic flows on the sphere and the projective plane. Their proof is based on an auxiliary lemma which, despite the validity of its statement, it is not well proved in the paper. I shall present how we found the problem in that paper (just the context, I will not bore you with technical details!) and (the idea of) how we solved it.
I would like to take advantage of this talk to propose a discussion about what proving in maths really is.

4 June 2015
Hendrik Weyer
(University of Bremen)
Spectral properties of measure-geometric Laplacians

Abstract:
In this talk we will discuss measure-geometric Laplacians $$\Delta^{\mu}$$ with respect to compactly supported atomless Borel probability measures $$\mu$$ as introduced by Freiberg and Z�hle in 2002. We will recall various basic properties and present recent results on the spectral properties of $$\Delta^{\mu}$$. Particularly we will classify the eigenfunctions of $$\Delta^{\mu}$$ under both Dirichlet and von Neumann boundary conditions. We will conclude by illustrating our results through several examples.
(This is joint work with M. Kesseb�hmer and T. Samuel)

### Academic Year 2014/2015 (Semester 1):

Date Speaker Title & Abstract
3 October 2014
Sascha Troscheit

Abstract:
In this talk I will give a short introduction to Assouad dimension and how it differs to other types of fractal dimension�. The main part of the talk will consist of introducing various models of random fractals and a brief survey of work done in the area. The talk will involve lots of handwaving and plenty of pictures and very little actual analysis.
(Joint work with Jonathan Fraser and Jun Jie Miao)
slides

10 October 2014 Tom Bourne
Multilingual Monoids: Why Study Varieties of Languages?

Abstract:
An introductory talk into why we should study the connections between families of monoids and families of languages, rather than individual monoids and individual languages. Variety theory will be introduced in order to highlight Eilenberg's Variety Theorem as the essential part of the framework. Examples throughout (hopefully)!
slides

17 October 2014
Artur Sch�fer
The central dogma of genetics: Or rather the coding theory behind it

Abstract:
The central dogma of genetics says, that the coded genetic information (DNA) is transcribed into transportable cassettes, composed of messenger RNA which contains the program for synthesis of a particular protein. In short: Information is encoded, sent through a channel and decoded into useful information. Having this highly interesting application as our background, we are going to take a look at the basics and non-basics of coding theory. Codes can be defined using as little algebraic structure as possible, but by providing more it might generate better codes. Thus, we will start from looking at basic codes, moving to linear and cyclic codes and afterwards presenting group ring codes.
Following that approach, the "amount" of algebraic structure is increasing, but does it always give us better codes?
And what are good and bad codes, actually?
slides

24 October 2014
Casey Donoven
Friendly Introduction to Fractal Dimension

Abstract:
I will introduction the Hausdorff dimension and give some motivation for its existence through examples and less intricate notions of dimension.  I might also describe the box-counting dimension and some multifractal theory.

31 October 2014
Michael Torpey
Computing with Semigroup Congruences

Abstract:
Congruences crop up everywhere in algebra, though we may not always know it. Any algebraic object has attached to it some theory which allows us to
partition it, take quotients and find homomorphic images.  In group theory we
use normal subgroups; in ring theory, two-sided ideals. In semigroup theory
in general, we can't do much better than study congruences directly, which is
very expensive if we want to do anything computational. However, after some
definitions, I will present a few different ways that we can abstract
semigroup congruences to other objects, and we will see how these abstractions
might help us compute things we want to know.

7 November 2014
Wilfred Wilson
Computing the maximal subsemigroups of finite semigroups

Abstract:
It is easy to ask (but much more difficult to answer) the question: given a
finite semigroup, what are its maximal subsemigroups? Fortunately there
is enough theory to allow us to compute these things in a practical way.
We firstly calculate the maximal subsemigroups of a few easy classes of
semigroup, and then proceed to discuss how the "MaximalSubsemigroups"
function works in the Semigroups package.

14 November 2014
Julius Jonusas
Approximations in automorphism groups

Abstract:
Short introduction to Fra�ss� limits and approximations of automorphisms on them.

21 November 2014
Feyisayo Olukoya
The conjugacy problem in groups

Abstract:
The conjuagacy (or transformation) problem is one of three group-theoretic decision problems proposed by Max Dehn in 1912 which he considered to be fundamental to group theory. The conjugacy problem has since been proven to be undecidable in general. There has been a lot of work done on the conjugacy problem, too much to be covered in the scope of our discussion and with many remarkable results, hence we shall be focusing only on some of these results. We begin by introducing the problem and considering a solution to the transformation problem in finite groups --- firstly the finite symmetric group and then in greater generality all finite groups.The finitely generated free groups will give us an example of an infinite group with solvable conjugacy problem. We then move on to look at insolubility cases, which will naturally force a more thorough examination of the definitions of algorithm, decidability, undecidability, semi-decidability solvability and insolubility, and how these arise in infinite groups. As a result of this, we are led to examine the question of the interplay between the conjugacy problem and certain familiar group-theoretic constructions. In this we shall be focused on extensions of groups and here we shall present the result which will form the highlight of the discussion time permitting.

This talk is based entirely on my final year project.

28 November 2014
Daniel Bennett
Demonstrative Embedding's in Thompson's Group V

Abstract:
There exists a conjecture in the area of CoCF groups, that was proposed in it's original form by Lehnert and revised by Bleak, Matucci and Neunhoffer, which proposes that Thompson's Group V is a Universal CoCF Group. We will explore what this means and steps that are being taken to prove/disprove this conjecture. An active area of study that looks promising in this regard are the demonstrative subgroups of V. The second half of the talk will be uncovering what these are, looking at the closure properties of these groups and ending with their relevance to the current conjecture.

5 December 2014
Gavin Abernethy
Normal numbers

Abstract:
I shall be discussing some ideas about normal and non-normal numbers, using results regarding the uniform distribution of sequences as well as from ergodic theory. This talk shall be based on a part of my senior honours project from last year.

12 December 2014
Christmas Lunch

### Academic Year 2013/2014 (Semester 2):

Date Speaker Title & Abstract
4 March 2014
Thomas Bourne
Minimal Automata and the Method of Quotients

Abstract:
We will explore how to algorithmically construct the minimal automaton for a regular language given some regular expression representing said language. The technique used to do this is the Method of Quotients. The theory will be demonstrated with an extended example.
11 March 2014
Abel Farkas
Projections and other images of self similar sets

Abstract:
It will be measure theory and dimension theory of mainly linear images of self-similar sets.
8 April 2014
Casey Donoven
Intersections within Cantor Spaces.

Abstract:
The codimension formula describes the dimension of the intersection of two fractal in R^n. I will explain my recent work with Kenneth on abstracting this result in Cantor Spaces.
15 April 2014
Sascha Troscheit
Recurrence and the renewal shift.

Abstract: In this talk I will describe the notion of recurrence in dynamical systems and give a combinatorical approach to the analysis of subsets of the shift space associated with the renewal shift with specified recurrence rates.
22 April 2014
Jenni Awang
Sorry we're closed, but that's a nice dress (aka Closures and Compl{i,e}ments)

Abstract:  A 1922 paper by Kuratowski described what happens when you take repeatedly take closures and complements of a set in a topological space.  We'll look at his result, and then swiftly move away from the world of topology and see how this can be applied to formal languages (Brzozowski, Grant, Shallit, 2009).
29 April 2014
Rachael Carey
Recognising semigroups

Abstract: Automatic semigroups, graph automatic semigroups and FA-presentable semigroups are all types of semigroups which can be represented by regular languages and recognised by finite state automata. In my talk I will define these concepts, outline some of the properties of each, and explain how they interact with one another.
6 May 2014
Artur Sch�fer
Synchronization and Graphendomorphisms

Abstract: In this talk we will introduce the term synchronizing permutation groups and find endomorphisms of graphs corresponding to non-synchronizing permutation groups. Only primitive groups can be synchronizing; therefore, we are able to apply the classification of O'Nan Scott of primitive groups and try to classify synchronizing groups. Also, groups can be attached to undirected graphs and the synchronization property can be translated into a graph property. Vaguely the theorem states: A group is synchronizing iff its graphs have no proper endomorphisms. We will apply this theorem to the Hamming graph and investigate the structure of its endomorphisms.
13 May 2014
Daniel Bennett
Context Free Grammars, Languages and Groups

Abstract: A brief introduction to Context Free Grammars and Languages. We will take the natural progression onto Context Free and CoCF groups, with examples of all these concepts throughout.
20 May 2014
Vuksan Mijovic Multifractal zeta functions

Abstract: Talk will be based on joint paper with Lars Olsen in which we introduce multifractal zeta-functions providing precise information of a very general class of multifractal spectra, including, for example, the multifractal spectra of self-conformal measures and the multifractal spectra of ergodic Birkhoff averages of continuous functions. More precisely, we prove that these and more general multifractal spectra equal the abscissae of convergence of the associated zeta functions.
27 May 2014

### Academic Year 2013/2014 (Semester 1):

Date Speaker Title & Abstract
10 October 2013
Casey Donoven
Groups of Homoeomorphisms of Cantor Sets

Abstract:
Self-similar Cantor sets allow for many interesting homeomorphism through exploiting their similar, disconnected, and symmetrical cylinders
17 October 2013
Integer Partitions and Bijections

Abstract:
Let p(n) denote the number of partitions of the natural number n. This will be our main object of discussion. I will present some elementary theory of partitions and show how one can obtain beautiful/useful results about p(n) using generating functions or in some cases explicit bijections! One can follow this talk just by having a first year mathematics (undergraduate) background and a love for counting =)
24 October 2013
Sam Baynes
Relative d-sequences

Abstract:
An introduction to relative d-sequences, which allow us to consider some sense of growth of direct powers of non-finitely generated semigroups.
31 October 2013
Sascha Troscheit
Non-differentiability of self-similar devil's staircases

Abstract: Devil's staircases (or Cantor functions) can be seen as the cumulative probability distribution function of a finite measure supported on a Cantor-like fractal. I will discuss some results concerning the existence of derivatives on those functions and their link to multifractal spectra.
7 November 2013
Jenni Awang
Semigroups, Cayley Graphs and Finite Presentability

Abstract:  We will begin by defining each of these three things - no prior knowledge required! We will then take a brief tour through how these are connected and whether we can gain information about one from information from the other - for example, if two semigroups have similar Cayley graphs, can we decide if finite presentability of one implies finite presentability of the other?
14 November 2013
James Hyde

21 November 2013
Anna Schroeder
A short journey into the world of symmetry

Abstract: Whether it is a bumble bee looking for food, the design of a wallpaper or even group theory, they all rely on some underlying concept of symmetry. In my talk I want to take you on a trip to explore various topics related to symmetry. In particular I want to briefly talk about symmetry in nature before moving on to the so called wallpaper groups which tell you how many different types of symmetry, namely 17, you can find on wallpapers. I also plan to introduce you to the symmetries of a graph which leads to the notion of an automorphism group of a graph which played an important part in the discovery of new sporadic simple groups.
28 November 2013
Malte Koch
Feller-Dynkin semigroups

Abstract: I want to tell you why these are my favourite semigroups.
5 December 2013
Tom Bourne
Inverse Semigroups and Inductive Groupoids

Abstract: Successful attempts to axiomatize the notion of a pseudogroup in the 1950s led to the introduction of two different mathematical structures, namely inverse semigroups and inductive groupoids. In this talk I will discuss the main steps in the proof of the Ehresmann-Schein-Nambooripad Theorem; a theorem which establishes two separate isomorphisms between the category with inverse semigroups as objects and the category with inductive groupoids as objects, where the choice of morphisms in each category determines the isomorphism.
12 December 2013
Alex McLeman
Cayley Automaton Semigroups

Abstract: I will tell you what a Cayley Automaton Semigroup is and give some basic results on them, before discussing some recent work on self-automaton semigroups and so-called Cayley chains obtained from iterating the construction. This will be an overview of all the work I have done in the last 3.5 years and hopefully won't end with me realising just how little I've done
18 December 2013
Christmas PPS

Bring along a christmas related maths fact/problem/theorem and we'll have a maths-christmas party!

### Academic Year 2012/2013 (Semester 2):

Date Speaker Title & Abstract
13 March 2013
Abel Farkas
Dense subgroups in orthogonal groups.

3 April 2013
James Hyde
Sierpinski rank and universal sequences.

10 April 2013
Anna Schroeder
A short introduction to the "periodic tables of finite group theory".

Abstract:
Similar to the periodic tables in chemistry the "ATLAS of Finite Groups" contains a lot of information about the building blocks of finite groups. In my talk I'm first going to give you a short introduction to representation theory before telling you a bit more about how to read one of the main reference books for finite group theorists.
17 April 2013
Julius Jonusas
Fraisse limits and topological generation.

1 May 2013
Casey Donoven
An Algebraic Take on the Self-Similarity of the Cantor Set.

Abstract:
I will discussing groups of homeomorphisms of the Cantor set that "respect" its self-similarity and their various constructions and properties. These groups include the Thompson Groups $$F$$, $$T$$, and $$V$$ and a new group I will call $$V_{ms}$$.
15 May 2013
Rachael Carey
Direct products of graph automatic semigroups.

Abstract: I will discuss how the property of graph automaticity is preserved when taking direct products.
22 May 2013
Alex McLeman
Introduction to Category Theory

Abstract:
Collin told me I should learn what a category is, so you're all going to share in my pain. I'll tell you what a category is and explain a few of their basic properties.

### Academic Year 2012/2013 (Semester 1):

Date Speaker Title & Abstract
10 October 2012
Lighting Talks
Jenni Awang - Semimetric Spaces
Jon Fraser - F�rstenberg's Topological Proof for the Infinitude of Primes
Alex McLeman - Theorem on Friends and Strangers (for larger groups)
Ruth Hoffmann - Pancake Sort
James Hyde - Sierpinski Rank of the Monoid of Partial Bijections
17 October 2012
Jenni Awang
The Fundamental Theorem of Geometric Group Theory (or how I learned to stop worrying and love the Svarc-Milnor Lemma.

Abstract:
Seemingly everything has a fundamental theorem - algebra, arithmetic, calculus, finitely generated abelian groups. The �varc-Milnor Lemma is sometimes referred as such a theorem for geometric group theory - roughly stating that if a group acts on a metric space in a nice way, then the group resembles the space. I will explain what we mean by "nice" and "resemble", and show how I have applied the lemma in my own research.
24 October 2012
Anna Schroeder
Finding Maximal Subgroups of Classical Groups

Abstract:
Many classical groups are simple, and by the Jordan-Hoelder Theorem, simple groups are the building blocks of all finite groups. In my talk I will define classical groups and explain why knowing their maximal subgroups can be useful. I will also talk about how one might go about finding them.
31 October 2012
Rachael Carey
Graph Automatic Semigroups.

Abstract:
I will introduce the concept of graph automatic semigroups and some of their basic properties.
7 November 2012
Sam Baynes
Introduction to Green's *-relations

28 November 2012
Jon Fraser
Generic dimensions of graphs and images

Abstract:
Working in the Banach space of continuous functions from an arbitrary compact metric space into n-dimensional Euclidean space, I will consider two questions: what is the generic dimension of the graph? and what is the generic dimension of the image? I will ask these questions in two different contexts, namely, Baire category and prevalence. Interestingly, these approaches will give starkly different answers to both questions.
5 December 2012
Alex McLeman
Self-Automaton Semigroups

Abstract:
I will describe what an automaton semigroup is and then move on to the specific case of Cayley Automaton Semigroups. After looking at some examples and properties will will discuss what it means for a semigroup to be self-automaton and attempt to classify some of them.
12 December 2012
Ruth Hoffmann
Decomposition Algorithms for Permutation Pattern Classes

19 December 2012
Christmas PPS