# Publications

### Publications

Aslan, B, Platt, D and Sheard, D (2023). Group invariant machine learning by fundamental domain projections. Proceedings of the 1st NeurIPS Workshop on Symmetry and Geometry in Neural Representations, in Proceedings of Machine Learning Research 197:181-218. Journal PDF

### Preprints

Classifying reflection generating sets of Coxeter groups (in preparation)

Quiver mutations and reflection equivalence in Weyl groups (in preparation)

Free completion sequences and Nielsen equivalence in RACGs (in preparation)

Nielsen equivalence invariants for Coxeter groups (in preparation)

Dynamical systems on orbifolds, joint work with d'Alfonso del Sordo, A (in preparation)

### Theses, projects reports, and dissertations

Sheard, D (2022). Nielsen equivalence in Coxeter groups; and a geometric approach to group equivariant machine learning. PhD thesis. Supervised by Larsen Louder. PDF (submitted version (2022)) PDF (current version with fewer typos (2023))

Sheard, D (2018). Palindromic Bruhat ideals and hyperplane arrangements. LSGNT mini project report. Supervised by Konstanze Rietsch. PDF

Sheard, D (2018). Introduction to the accessibility of groups. LSGNT mini project report. Supervised by Larsen Louder. PDF

Sheard, D (2017). The many complexities of Coxeter groups. MMath dissertation. Supervised by Pavel Tumarkin. PDF

### 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.

### GEOMETRIC APROACHES TO NIELSEN EQUIVALENCE

Oxford University junior topology and group theory seminar 10th March 2021

UCL/KCL junior geometry seminar 4th February 2021

In the world of finitely generated groups, presentations are a blessing and a curse. They are versatile and compact, but in general tell you very little about the group. Tietze transformations offer much, but deliver little, in terms of understanding the possible presentations of a group. I will introduce a different way of transforming presentations of a group called a Nielsen transformation, and show how geometric methods can be used to study Nielsen transformations.

### GROUP INVARIANT MACHINE LEARNING VIA GEOMETRIC TECHNIQUES

Imperial College London junior geometry seminar 19th February 2021

Applications of machine learning have become ubiquitous in our everyday lives, and are steadily becoming more common even in basic research. In a nutshell, machine learning seeks to use computers to approximate highly complex, analytically intractable functions, with simple functions; such a function might map the intersection matrix of a complete intersection Calabi–Yau manifold to its Hodge number, for example.

In this talk I will discuss joint work with Ben Aslan and Daniel Platt, in which we apply methods from differential geometry and geometric group theory to design more efficient and accurate machine learning algorithms in the cases where the function to be approximated is invariant under the action of some group.

### 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.