Teaching

I taught a mini-course on axial algebras to the staff and postgrads in May 2018. Before that, I taught part of the Topics in Discrete Mathematics course at Bristol in 2014-15, details of which can be found below. The details of my previous teaching can be found here.

Introduction to axial algebras

I gave an invited lecture series on axial algebras at the CIMPA School: Non-associative Algebras and their Applications in Madagascar (online) from 30th August to 11th September 2021. The course notes can be found here and the slides for the final talk here.

Code algebras, axial algebras and VOAs

I gave an invited lecture series at the CIMPA-African Mathematical School: Algebras, Geometries and Permutation Groups in Durban in December 2019.

Axial algebras mini-course

I taught a short course on axial algebras to the staff and postgrads in May 2018. This was an introductory course aimed at getting people prepared for the 5-day workshop I organised on axial algebras which was in May/June 2018. It should give the reader some motivation, a good background in the subject and an overview of axial algebras. The lecture notes can be found here.

Topics in Discrete Mathematics - Permutation groups

I am teaching the second of three parts of the 3rd year Topics in Discrete Mathematics course. This part will be on permutation groups. The first part will be on graph theory and will be taught by Tom Bloom; the third is on designs and will be taught by Jason Semeraro.

Originally, permutation groups meant subgroups of the symmetric group $S_n$. However, we will see that all groups can be embedded as a subgroup of a permutation group. So, just as $S_n$ acts naturally on the set $\{1, \dots, n\}$, we will define and study the action of a group on some object. Such a situation occurs naturally by taking the automorphism group of a graph, a design, or some other combinatorial object. We will cover the orbit-stabiliser theorem and learn about primitive and imprimitive actions. We will prove some results which demonstrate that primitive groups are small and rare. We will also briefly look at multiply transitive groups and we will show that these are also rare.

The complete lecture notes are below - please let me know if you find any typos. Although the exercise sheets do not count towards your final mark, you are strongly encouraged to do them to help your understanding of the course.

• Lecture notes -- Complete, updated 20/02/15


• Exercise sheet 1


• Solutions 1


• Exercise sheet 2


• Solutions 2 -- updated qn4 10/5/15

My office hour is 3pm on Fridays straight after the Friday lecture in 2.22 Howard House.

Projects

• The O'Nan Scott Theorem - The original statement of this theorem was classifying the possible maximal subgroups of $S_n$ and $A_n$. However, Peter Cameron noticed that using the Classification of Finite Simple Groups it could be restated to restrict the structure of primitive groups into one of a few different types. This is usually how the theorem is now stated. You could either describe and prove the original theorem and discuss how it can be applied to primitive groups, or discuss the restated theorem and the different types of primitive group given in it. This project would be more suitable for someone with a strong background in group theory.

• Orbitals and primitive groups - If $G$ acts transitively on $\Omega$, one can consider the action of $G$ on the cartesian product $\Omega \times \Omega$. The orbits of such an action are called orbitals. If $\Delta$ is such an orbital then we can define a graph with vertex set $\Omega$ and edges given by $\Delta$. You would explore the connections between this graph, orbitals and the structure of the permutation group.

Previous teaching

Whilst at Leicester, I lectured the third year course Curves and Surfaces - my notes for the 2012/13 course can be found here. That year, I also lectured the first year course on Euclidean geometry and the third year reading course on Knot theory.

At Lincoln College, Oxford, I lectured in college to the first and second years, covering real and complex analysis and measure theory.

I have also taught a large variety of example classes across pure mathematics (and sometimes beyond!).