# Publications

Here are some selected publications

### Preprints

Nielsen equivalence in Coxeter groups (in preparation).

Group invariant machine learning through near-isometries, joint with Benjamin Aslan and Daniel Platt (in preparation)

### Posters

Group invariant machine learning through near-isometries, joint with Benjamin Aslan and Daniel Platt, exhibited as part of the Data Science Virtual Poster Competition 2020 hosted by Imperial College London.

## TALKS

Here is a list of some academic talks I have given in the past.

### CUTTING UP SURFACES AND COMMUTATORS IN FREE GROUPS

Warwick DAGGER seminar 13th Jan 2020

Bristol junior geometry seminar 5th Nov 2019

UCL/KCL junior geometry seminar 10th Oct 2019

When is an element g in a group G a product of commutators [a,b]? What is the minimal number of commutators n such that g=[a_1,b_1]...[a_n,b_n]? What are all possible solutions (a_1,b_1,...,a_m,b_m) to the equation

g=[a_1,b_1]...[a_m,b_m]? Very difficult, yet important, questions - but ones whose answers seem to lie in the darkest recesses of combinatorial group theory. Not so! At least for G a free group, these questions can all be answered elegantly and beautifully by cutting up and colouring surfaces. In this talk I shall present solutions to these problems with an emphasis on drawing nice pictures.

### THE TOPOLOGY OF KNOTS AND SURFACES

UCL undergraduate colloquium 13th Nov 2019

Topology, or the mathematical study of the shape of things, is one of the most important areas of mathematics, underpinning, aiding, or influencing a diverse range of modern mathematics including geometry, number theory, combinatorics and group theory. It even has applications to quantum mechanics, molecular biology, and computer science. This talk will be a whistle stop tour of some 1 and 2 dimensional topology, including knots and links, Seifert surfaces, polygonal complexes and Euler characteristic, and finishing off with the classification of compact surfaces. - Oh, and the best part? There will be lots of pictures!

### CAN FORGETFUL COMPUTERS DO GROUP THEORY?

UCL postgraduate seminar 25th Apr 2019

In 1931 Gödel proved that there were problems in mathematics which can never be solved...fiddlesticks! To add insult to injury in 1936 Church and Turing independently showed that even among those problems which are solvable, there are some which take an infinite amount of time to solve algorithmically...double-fiddlesticks!! Since this time, questions of computability (decision problems) have entered all corners of mathematics, some of the most well-known being posed by Dehn: the word, conjugacy, and isomorphism problems in group theory. I shall introduce a model for a particularly stupid type of computer (a Turing machine with no memory) called a finite state automaton, and discuss whether such computers can do group theory. If/when they can, I'll discuss how they can help solve some of Dehn's decision problems.

### AN INTRODUCTION TO GEOMETRIC GROUP THEORY WITH APPLICATIONS TO ACCESSIBILITY

UCL/KCL junior geometry seminar 8th Nov 2018

Geometric group theory aims to study groups using tools from geometry and topology, more precisely to study finitely generated groups by looking at the way they act on geometric and topological spaces. In this talk I shall introduce some of the aims of geometric group theory, and construct a well-known space associated to a finitely presented group. I will talk about algebraic splittings of groups in the sense of van Kampen's Theorem, and finish by sketching the proof of Dunwoody's theorem that finitely presented groups admit only a finite number of such splittings over finite subgroups.

### THE DUALITY BETWEEN THE COMBINATORICS AND THE GEOMETRY OF COXETER GROUPS

UCL/KCL junior geometry seminar 2nd Nov 2017

Coxeter groups, or groups generated by reflections which act discretely, arise naturally as the symmetry groups of regular polytopes and periodic tilings. They are typically studied from one of two viewpoints. From the point of view of geometry they give rise to nice simplicial structures which are very important in the study of Lie groups and Lie algebras. On the other hand, thought of as combinatorial groups, they admit a simple constructive solution to the word problem, as well as a solution to the conjugacy problem. I will introduce the basics of both of these approaches and show how the algebraic structure of the group is encoded into the geometry, and illustrate how this can dramatically simplify proofs of combinatorial results, as well as giving us a geometric solution to the word problem.